Analysis I, Section 11.5: Riemann integrability of continuous functions

I have attempted to make the translation as faithful a paraphrasing as possible of the original text. When there is a choice between a more idiomatic Lean solution and a more faithful translation, I have generally chosen the latter. In particular, there will be places where the Lean code could be "golfed" to be more elegant and idiomatic, but I have consciously avoided doing so.

Main constructions and results of this section:

namespace Chapter11open BoundedIntervalopen Chapter9

Theorem 11.5.1

theorem declaration uses 'sorry'integ_of_uniform_cts {I: BoundedInterval} {f:ℝ → ℝ} (hf: UniformContinuousOn f I) : IntegrableOn f I := I:BoundedIntervalf:ℝ → ℝhf:UniformContinuousOn f ↑I⊢ IntegrableOn f I -- This proof is written to follow the structure of the original text. have hfbound : BddOn f I := I:BoundedIntervalf:ℝ → ℝhf:UniformContinuousOn f ↑I⊢ IntegrableOn f I I:BoundedIntervalf:ℝ → ℝhf:UniformContinuousOn f ↑I⊢ Bornology.IsBounded (f '' ↑I); All goals completed! 🐙 I:BoundedIntervalf:ℝ → ℝhf:UniformContinuousOn f ↑Ihfbound:Chapter9.BddOn _fvar.237 ↑_fvar.236 := ?_mvar.328⊢ lower_integral f I = upper_integral f I I:BoundedIntervalf:ℝ → ℝhf:UniformContinuousOn f ↑Ihfbound:Chapter9.BddOn _fvar.237 ↑_fvar.236 := ?_mvar.328hsing:I.length = 0⊢ lower_integral f I = upper_integral f II:BoundedIntervalf:ℝ → ℝhf:UniformContinuousOn f ↑Ihfbound:Chapter9.BddOn _fvar.237 ↑_fvar.236 := ?_mvar.328hsing:¬I.length = 0⊢ lower_integral f I = upper_integral f I I:BoundedIntervalf:ℝ → ℝhf:UniformContinuousOn f ↑Ihfbound:Chapter9.BddOn _fvar.237 ↑_fvar.236 := ?_mvar.328hsing:I.length = 0⊢ lower_integral f I = upper_integral f I All goals completed! 🐙 I:BoundedIntervalf:ℝ → ℝhf:UniformContinuousOn f ↑Ihfbound:Chapter9.BddOn _fvar.237 ↑_fvar.236 := ?_mvar.328hsing:I.a < I.b⊢ lower_integral f I = upper_integral f I I:BoundedIntervalf:ℝ → ℝhf:UniformContinuousOn f ↑Ihfbound:Chapter9.BddOn _fvar.237 ↑_fvar.236 := ?_mvar.328a:ℝ := Chapter11.BoundedInterval.a _fvar.236hsing:a < I.b⊢ lower_integral f I = upper_integral f I I:BoundedIntervalf:ℝ → ℝhf:UniformContinuousOn f ↑Ihfbound:Chapter9.BddOn _fvar.237 ↑_fvar.236 := ?_mvar.328a:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < b⊢ lower_integral f I = upper_integral f I have hsing' : 0 < b-a := I:BoundedIntervalf:ℝ → ℝhf:UniformContinuousOn f ↑I⊢ IntegrableOn f I All goals completed! 🐙 have (ε:ℝ) (hε: ε > 0) : upper_integral f I - lower_integral f I ≤ ε * (b-a) := I:BoundedIntervalf:ℝ → ℝhf:UniformContinuousOn f ↑I⊢ IntegrableOn f I I:BoundedIntervalf:ℝ → ℝhf:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0⊢ upper_integral f I - lower_integral f I ≤ ε * (b - a) I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)⊢ upper_integral f I - lower_integral f I ≤ ε * (b - a); I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ ε⊢ upper_integral f I - lower_integral f I ≤ ε * (b - a) I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑N⊢ upper_integral f I - lower_integral f I ≤ ε * (b - a) have hNpos : 0 < N := I:BoundedIntervalf:ℝ → ℝhf:UniformContinuousOn f ↑I⊢ IntegrableOn f I have : 0 < (b-a)/δ := I:BoundedIntervalf:ℝ → ℝhf:UniformContinuousOn f ↑I⊢ IntegrableOn f I All goals completed! 🐙 I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑Nthis:0 < (_fvar.6554 - _fvar.6553) / _fvar.6577 := ?_mvar.13640⊢ 0 < ↑N; All goals completed! 🐙 have hN' : (b-a)/N < δ := I:BoundedIntervalf:ℝ → ℝhf:UniformContinuousOn f ↑I⊢ IntegrableOn f I rwa [div_lt_comm₀I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := ?_mvar.13528⊢ (b - a) / δ < ↑NI:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := ?_mvar.13528⊢ 0 < ↑NI:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := ?_mvar.13528⊢ 0 < δ I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := ?_mvar.13528⊢ 0 < ↑NI:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := ?_mvar.13528⊢ 0 < δ All goals completed! 🐙 have : ∃ P: Partition I, P.intervals.card = N ∧ ∀ J ∈ P.intervals, |J|ₗ = (b-a) / N := I:BoundedIntervalf:ℝ → ℝhf:UniformContinuousOn f ↑I⊢ IntegrableOn f I All goals completed! 🐙 I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := ?_mvar.13528hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := ?_mvar.21158P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑N⊢ upper_integral f I - lower_integral f I ≤ ε * (b - a) calc _ ≤ ∑ J ∈ P.intervals, (sSup (f '' J) - sInf (f '' J)) * |J|ₗ := I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑N⊢ upper_integral f I - lower_integral f I ≤ ∑ J ∈ P.intervals, (sSup (f '' ↑J) - sInf (f '' ↑J)) * J.length I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑Nh1:?_mvar.27058 := Chapter11.upper_integ_le_upper_sum _fvar.6552 _fvar.26458⊢ upper_integral f I - lower_integral f I ≤ ∑ J ∈ P.intervals, (sSup (f '' ↑J) - sInf (f '' ↑J)) * J.length I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑Nh1:?_mvar.27058 := Chapter11.upper_integ_le_upper_sum _fvar.6552 _fvar.26458h2:?_mvar.27072 := Chapter11.lower_integ_ge_lower_sum _fvar.6552 _fvar.26458⊢ upper_integral f I - lower_integral f I ≤ ∑ J ∈ P.intervals, (sSup (f '' ↑J) - sInf (f '' ↑J)) * J.length I:BoundedIntervalf:ℝ → ℝhfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bε:ℝδ:ℝhf✝:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑Nh1:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := Chapter11.upper_integ_le_upper_sum _fvar.6552 _fvar.26458h2:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := Chapter11.lower_integ_ge_lower_sum _fvar.6552 _fvar.26458hf:∀ (ε : ℝ), 0 < ε → ∃ δ, 0 < δ ∧ ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, dist x x₀ ≤ δ → dist (f x) (f x₀) ≤ εhsing':a < bhε:0 < εhδ:0 < δ⊢ upper_integral f I ≤ ∑ x ∈ P.intervals, sSup (f '' ↑x) * x.length - ∑ x ∈ P.intervals, sInf (f '' ↑x) * x.length + lower_integral f I All goals completed! 🐙 _ ≤ ∑ J ∈ P.intervals, ε * |J|ₗ := I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑N⊢ ∑ J ∈ P.intervals, (sSup (f '' ↑J) - sInf (f '' ↑J)) * J.length ≤ ∑ J ∈ P.intervals, ε * J.length I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑N⊢ ∀ i ∈ P.intervals, (sSup (f '' ↑i) - sInf (f '' ↑i)) * i.length ≤ ε * i.length; I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑NJ:BoundedIntervalhJ:J ∈ P.intervals⊢ (sSup (f '' ↑J) - sInf (f '' ↑J)) * J.length ≤ ε * J.length; I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑NJ:BoundedIntervalhJ:J ∈ P.intervals⊢ sSup (f '' ↑J) - sInf (f '' ↑J) ≤ ε have {x y:ℝ} (hx: x ∈ J) (hy: y ∈ J) : f x ≤ f y + ε := I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑N⊢ ∑ J ∈ P.intervals, (sSup (f '' ↑J) - sInf (f '' ↑J)) * J.length ≤ ∑ J ∈ P.intervals, ε * J.length I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑NJ:BoundedIntervalhJ:J ∈ P.intervalsx:ℝy:ℝhx:x ∈ Jhy:y ∈ Jthis:_fvar.94592 ⊆ _fvar.236 := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)⊢ f x ≤ f y + ε have : |f x - f y| ≤ ε := I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑N⊢ ∑ J ∈ P.intervals, (sSup (f '' ↑J) - sInf (f '' ↑J)) * J.length ≤ ∑ J ∈ P.intervals, ε * J.length I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑NJ:BoundedIntervalhJ:J ∈ P.intervalsx:ℝy:ℝhx:x ∈ Jhy:y ∈ Jthis:_fvar.94592 ⊆ _fvar.236 := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)⊢ y ∈ ↑II:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑NJ:BoundedIntervalhJ:J ∈ P.intervalsx:ℝy:ℝhx:x ∈ Jhy:y ∈ Jthis:_fvar.94592 ⊆ _fvar.236 := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)⊢ x ∈ ↑II:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑NJ:BoundedIntervalhJ:J ∈ P.intervalsx:ℝy:ℝhx:x ∈ Jhy:y ∈ Jthis:_fvar.94592 ⊆ _fvar.236 := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)⊢ |x - y| ≤ δ I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑NJ:BoundedIntervalhJ:J ∈ P.intervalsx:ℝy:ℝhx:x ∈ Jhy:y ∈ Jthis:_fvar.94592 ⊆ _fvar.236 := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)⊢ y ∈ ↑II:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑NJ:BoundedIntervalhJ:J ∈ P.intervalsx:ℝy:ℝhx:x ∈ Jhy:y ∈ Jthis:_fvar.94592 ⊆ _fvar.236 := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)⊢ x ∈ ↑II:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑NJ:BoundedIntervalhJ:J ∈ P.intervalsx:ℝy:ℝhx:x ∈ Jhy:y ∈ Jthis:_fvar.94592 ⊆ _fvar.236 := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)⊢ |x - y| ≤ δ try All goals completed! 🐙 I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑NJ:BoundedIntervalhJ:J ∈ P.intervalsx:ℝy:ℝhx:x ∈ Jhy:y ∈ Jthis:_fvar.94592 ⊆ _fvar.236 := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)⊢ J.length ≤ δ; All goals completed! 🐙 All goals completed! 🐙 have hJnon : (f '' J).Nonempty := I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑N⊢ ∑ J ∈ P.intervals, (sSup (f '' ↑J) - sInf (f '' ↑J)) * J.length ≤ ∑ J ∈ P.intervals, ε * J.length I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑NJ:BoundedIntervalhJ:J ∈ P.intervalsthis:∀ {x y : ℝ}, x ∈ _fvar.94592 → y ∈ _fvar.94592 → @_fvar.237 x ≤ @_fvar.237 y + _fvar.6557 := fun {x y} hx hy => @?_mvar.108668 x y hx hy⊢ (↑J).Nonempty; I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑NJ:BoundedIntervalhJ:J ∈ P.intervalsthis:∀ {x y : ℝ}, x ∈ _fvar.94592 → y ∈ _fvar.94592 → @_fvar.237 x ≤ @_fvar.237 y + _fvar.6557 := fun {x y} hx hy => @?_mvar.108668 x y hx hyh:↑J = ∅⊢ False replace h : Subsingleton (J:Set ℝ) := I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑N⊢ ∑ J ∈ P.intervals, (sSup (f '' ↑J) - sInf (f '' ↑J)) * J.length ≤ ∑ J ∈ P.intervals, ε * J.length All goals completed! 🐙 I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑NJ:BoundedIntervalhJ:J ∈ P.intervalsthis:∀ {x y : ℝ}, x ∈ _fvar.94592 → y ∈ _fvar.94592 → @_fvar.237 x ≤ @_fvar.237 y + _fvar.6557 := fun {x y} hx hy => @?_mvar.108668 x y hx hyh:(b - a) / ↑N = 0⊢ False linarith [show 0 < (b-a) / N I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑N⊢ ∑ J ∈ P.intervals, (sSup (f '' ↑J) - sInf (f '' ↑J)) * J.length ≤ ∑ J ∈ P.intervals, ε * J.length All goals completed! 🐙] replace (y:ℝ) (hy:y ∈ J) : sSup (f '' J) ≤ f y + ε := I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑N⊢ ∑ J ∈ P.intervals, (sSup (f '' ↑J) - sInf (f '' ↑J)) * J.length ≤ ∑ J ∈ P.intervals, ε * J.length I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑NJ:BoundedIntervalhJ:J ∈ P.intervalsthis:∀ {x y : ℝ}, x ∈ _fvar.94592 → y ∈ _fvar.94592 → @_fvar.237 x ≤ @_fvar.237 y + _fvar.6557 := fun {x y} hx hy => @?_mvar.108668 x y hx hyhJnon:(_fvar.237 '' ↑_fvar.94592).Nonempty := ?_mvar.126486y:ℝhy:y ∈ J⊢ ∀ b ∈ f '' ↑J, b ≤ f y + ε; All goals completed! 🐙 replace : sSup (f '' J) - ε ≤ sInf (f '' J) := I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑N⊢ ∑ J ∈ P.intervals, (sSup (f '' ↑J) - sInf (f '' ↑J)) * J.length ≤ ∑ J ∈ P.intervals, ε * J.length I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑NJ:BoundedIntervalhJ:J ∈ P.intervalshJnon:(_fvar.237 '' ↑_fvar.94592).Nonempty := ?_mvar.126486this:∀ y ∈ _fvar.94592, sSup (_fvar.237 '' ↑_fvar.94592) ≤ @_fvar.237 y + _fvar.6557 := fun y hy => @?_mvar.135805 y hy⊢ ∀ b ∈ f '' ↑J, sSup (f '' ↑J) - ε ≤ b; All goals completed! 🐙 All goals completed! 🐙 _ = ∑ J ∈ P.intervals, ε * (b-a)/N := I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑N⊢ ∑ J ∈ P.intervals, ε * J.length = ∑ J ∈ P.intervals, ε * (b - a) / ↑N All goals completed! 🐙 _ = _ := I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑N⊢ ∑ J ∈ P.intervals, ε * (b - a) / ↑N = ε * (b - a) I:BoundedIntervalf:ℝ → ℝhf✝:∀ ε > 0, ∃ δ > 0, ∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, δ.Close x x₀ → ε.Close (f x) (f x₀)hfbound:BddOn f ↑Ia:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < b - aε:ℝhε:ε > 0δ:ℝhδ:δ > 0hf:∀ x₀ ∈ ↑I, ∀ x ∈ ↑I, |x - x₀| ≤ δ → |f x - f x₀| ≤ εN:ℕhN:(b - a) / δ < ↑NhNpos:0 < _fvar.13474 := have this := div_pos _fvar.6556 _fvar.6585; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 _fvar.13474) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑_fvar.13474) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑_fvar.13474)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast _fvar.13474)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑_fvar.13474)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast _fvar.13474)))))) (Decidable.byContradiction fun a => ne_of_lt _fvar.13477 (le_antisymm (le_trans (le_of_lt _fvar.13477) (le_refl ↑_fvar.13474)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((_fvar.6554 - _fvar.6553) / _fvar.6577))))))hN':(_fvar.6554 - _fvar.6553) / ↑_fvar.13474 < _fvar.6577 := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr _fvar.13529) _fvar.6585)))) _fvar.13477P:Partition Ihcard:P.intervals.card = Nhlength:∀ J ∈ P.intervals, J.length = (b - a) / ↑N⊢ ↑N * (ε * (b - a) / ↑N) = ε * (b - a); All goals completed! 🐙 have lower_le_upper : 0 ≤ upper_integral f I - lower_integral f I := I:BoundedIntervalf:ℝ → ℝhf:UniformContinuousOn f ↑I⊢ IntegrableOn f I All goals completed! 🐙 I:BoundedIntervalf:ℝ → ℝhf:UniformContinuousOn f ↑Ihfbound:Chapter9.BddOn _fvar.237 ↑_fvar.236 := ?_mvar.328a:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < _fvar.5638 - _fvar.5555 := ?_mvar.5827this:∀ ε > 0, Chapter11.upper_integral _fvar.237 _fvar.236 - Chapter11.lower_integral _fvar.237 _fvar.236 ≤ ε * (_fvar.5638 - _fvar.5555) := fun ε hε => @?_mvar.6516 ε hεlower_le_upper:0 ≤ Chapter11.upper_integral _fvar.237 _fvar.236 - Chapter11.lower_integral _fvar.237 _fvar.236 := ?_mvar.179829h:0 < upper_integral f I - lower_integral f I⊢ lower_integral f I = upper_integral f II:BoundedIntervalf:ℝ → ℝhf:UniformContinuousOn f ↑Ihfbound:Chapter9.BddOn _fvar.237 ↑_fvar.236 := ?_mvar.328a:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < _fvar.5638 - _fvar.5555 := ?_mvar.5827this:∀ ε > 0, Chapter11.upper_integral _fvar.237 _fvar.236 - Chapter11.lower_integral _fvar.237 _fvar.236 ≤ ε * (_fvar.5638 - _fvar.5555) := fun ε hε => @?_mvar.6516 ε hεlower_le_upper:0 ≤ Chapter11.upper_integral _fvar.237 _fvar.236 - Chapter11.lower_integral _fvar.237 _fvar.236 := ?_mvar.179829h:0 = upper_integral f I - lower_integral f I⊢ lower_integral f I = upper_integral f I I:BoundedIntervalf:ℝ → ℝhf:UniformContinuousOn f ↑Ihfbound:Chapter9.BddOn _fvar.237 ↑_fvar.236 := ?_mvar.328a:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < _fvar.5638 - _fvar.5555 := ?_mvar.5827this:∀ ε > 0, Chapter11.upper_integral _fvar.237 _fvar.236 - Chapter11.lower_integral _fvar.237 _fvar.236 ≤ ε * (_fvar.5638 - _fvar.5555) := fun ε hε => @?_mvar.6516 ε hεlower_le_upper:0 ≤ Chapter11.upper_integral _fvar.237 _fvar.236 - Chapter11.lower_integral _fvar.237 _fvar.236 := ?_mvar.179829h:0 < upper_integral f I - lower_integral f I⊢ lower_integral f I = upper_integral f I I:BoundedIntervalf:ℝ → ℝhf:UniformContinuousOn f ↑Ihfbound:Chapter9.BddOn _fvar.237 ↑_fvar.236 := ?_mvar.328a:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < _fvar.5638 - _fvar.5555 := ?_mvar.5827this:∀ ε > 0, Chapter11.upper_integral _fvar.237 _fvar.236 - Chapter11.lower_integral _fvar.237 _fvar.236 ≤ ε * (_fvar.5638 - _fvar.5555) := fun ε hε => @?_mvar.6516 ε hεlower_le_upper:0 ≤ Chapter11.upper_integral _fvar.237 _fvar.236 - Chapter11.lower_integral _fvar.237 _fvar.236 := ?_mvar.179829h:0 < upper_integral f I - lower_integral f Iε:ℝ := (Chapter11.upper_integral _fvar.237 _fvar.236 - Chapter11.lower_integral _fvar.237 _fvar.236) / (2 * (_fvar.5638 - _fvar.5555))⊢ lower_integral f I = upper_integral f I replace : upper_integral f I - lower_integral f I ≤ (upper_integral f I - lower_integral f I)/2 := I:BoundedIntervalf:ℝ → ℝhf:UniformContinuousOn f ↑I⊢ IntegrableOn f I convert this ε (I:BoundedIntervalf:ℝ → ℝhf:UniformContinuousOn f ↑Ihfbound:Chapter9.BddOn _fvar.237 ↑_fvar.236 := Eq.mpr (id (congrArg (fun _a => _a) (propext (Chapter9.BddOn.iff' _fvar.237 ↑_fvar.236)))) (Chapter9.UniformContinuousOn.of_bounded _fvar.238 (@subset_rfl (Set ℝ) Set.instHasSubset (↑_fvar.236) Set.instIsReflSubset) (Chapter11.Bornology.IsBounded.of_boundedInterval _fvar.236))a:ℝ := Chapter11.BoundedInterval.a _fvar.236b:ℝ := Chapter11.BoundedInterval.b _fvar.236hsing:a < bhsing':0 < _fvar.5638 - _fvar.5555 := lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.5555) (Mathlib.Tactic.Ring.atom_pf _fvar.5638) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.5638 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.5555 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.5638 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.5638) (Mathlib.Tactic.Ring.atom_pf _fvar.5555) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.5555 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (_fvar.5555 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.5638 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.5555 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (_fvar.5638 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.5555 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.5638 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.5679) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)))this:∀ ε > 0, Chapter11.upper_integral _fvar.237 _fvar.236 - Chapter11.lower_integral _fvar.237 _fvar.236 ≤ ε * (_fvar.5638 - _fvar.5555) := fun ε hε => (fun δ x => (fun N hN => have hNpos := have this := div_pos _fvar.5828 x.left; Eq.mpr (id (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 N) (Eq.trans (Eq.trans (congrArg (fun x => x < ↑N) Nat.cast_zero) (Mathlib.Tactic.Qify.intCast_lt._simp_1 0 ↑N)) (Eq.trans (Eq.trans (congr (congrArg LT.lt Int.cast_zero) (Int.cast_natCast N)) (Mathlib.Tactic.Rify.ratCast_lt._simp_1 0 ↑N)) (congr (congrArg LT.lt Rat.cast_zero) (Rat.cast_natCast N)))))) (Decidable.byContradiction fun a => ne_of_lt hN (le_antisymm (le_trans (le_of_lt hN) (le_refl ↑N)) (le_trans (le_of_not_gt a) (le_trans (le_of_lt this) (le_refl ((Chapter11.BoundedInterval.b _fvar.236 - Chapter11.BoundedInterval.a _fvar.236) / δ)))))); have hN' := Eq.mpr (id (congrArg (fun _a => _a) (propext (div_lt_comm₀ (Nat.cast_pos'.mpr hNpos) x.left)))) hN; have this := sorry; (fun P x_1 => Trans.trans (Trans.trans (Trans.trans (have h1 := Chapter11.upper_integ_le_upper_sum _fvar.329 P; have h2 := Chapter11.lower_integ_ge_lower_sum _fvar.329 P; Eq.mpr (id (Eq.trans (Eq.trans (congrArg (LE.le (Chapter11.upper_integral _fvar.237 _fvar.236 - Chapter11.lower_integral _fvar.237 _fvar.236)) (Eq.trans (Finset.sum_congr (Eq.refl P.intervals) fun x a => sub_mul (sSup (_fvar.237 '' ↑x)) (sInf (_fvar.237 '' ↑x)) x.length) (Finset.sum_sub_distrib (fun x => sSup (_fvar.237 '' ↑x) * x.length) fun x => sInf (_fvar.237 '' ↑x) * x.length))) tsub_le_iff_right._simp_1) ge_iff_le._simp_1)) (le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (Chapter11.upper_integral _fvar.237 _fvar.236)) (Mathlib.Tactic.Ring.atom_pf (∑ J ∈ P.intervals, sSup (_fvar.237 '' ↑J) * J.length)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (∑ J ∈ P.intervals, sSup (_fvar.237 '' ↑J) * J.length) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Chapter11.upper_integral _fvar.237 _fvar.236 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((∑ J ∈ P.intervals, sSup (_fvar.237 '' ↑J) * J.length) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (∑ J ∈ P.intervals, sInf (_fvar.237 '' ↑J) * J.length)) (Mathlib.Tactic.Ring.atom_pf (Chapter11.lower_integral _fvar.237 _fvar.236)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (Chapter11.lower_integral _fvar.237 _fvar.236) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt ((∑ J ∈ P.intervals, sInf (_fvar.237 '' ↑J) * J.length) ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (Chapter11.lower_integral _fvar.237 _fvar.236 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Chapter11.upper_integral _fvar.237 _fvar.236 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt ((∑ J ∈ P.intervals, sSup (_fvar.237 '' ↑J) * J.length) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_zero_add ((∑ J ∈ P.intervals, sInf (_fvar.237 '' ↑J) * J.length) ^ Nat.rawCast 1 * Nat.rawCast 1 + (Chapter11.lower_integral _fvar.237 _fvar.236 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (∑ x ∈ P.intervals, sSup (_fvar.237 '' ↑x) * x.length)) (Mathlib.Tactic.Ring.atom_pf (∑ x ∈ P.intervals, sInf (_fvar.237 '' ↑x) * x.length)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (∑ J ∈ P.intervals, sInf (_fvar.237 '' ↑J) * J.length) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt ((∑ J ∈ P.intervals, sSup (_fvar.237 '' ↑J) * J.length) ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((∑ J ∈ P.intervals, sInf (_fvar.237 '' ↑J) * J.length) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.atom_pf (Chapter11.lower_integral _fvar.237 _fvar.236)) (Mathlib.Tactic.Ring.add_pf_add_lt ((∑ J ∈ P.intervals, sSup (_fvar.237 '' ↑J) * J.length) ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt ((∑ J ∈ P.intervals, sInf (_fvar.237 '' ↑J) * J.length) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_zero_add (Chapter11.lower_integral _fvar.237 _fvar.236 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.atom_pf (Chapter11.upper_integral _fvar.237 _fvar.236)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (Chapter11.upper_integral _fvar.237 _fvar.236) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Chapter11.upper_integral _fvar.237 _fvar.236 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_zero ((∑ J ∈ P.intervals, sSup (_fvar.237 '' ↑J) * J.length) ^ Nat.rawCast 1 * Nat.rawCast 1 + ((∑ J ∈ P.intervals, sInf (_fvar.237 '' ↑J) * J.length) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (Chapter11.lower_integral _fvar.237 _fvar.236 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (Chapter11.upper_integral _fvar.237 _fvar.236) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (∑ J ∈ P.intervals, sSup (_fvar.237 '' ↑J) * J.length) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (∑ J ∈ P.intervals, sInf (_fvar.237 '' ↑J) * J.length) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (Chapter11.lower_integral _fvar.237 _fvar.236) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_le_of_neg (Mathlib.Tactic.Linarith.add_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le h1) (Mathlib.Tactic.Linarith.sub_nonpos_of_le h2)) (Mathlib.Tactic.Linarith.sub_neg_of_lt a))))) (Finset.sum_le_sum fun J hJ => mul_le_mul_of_nonneg_right (have this := fun {x_2 y} hx hy => have this := P.contains _fvar.236 J hJ; have this_1 := x.right y (@this y hy) x_2 (@this x_2 hx) (LE.le.trans (Chapter11.BoundedInterval.dist_le_length hx hy) (Chapter11.integ_of_uniform_cts._proof_3 _fvar.329 _fvar.5679 _fvar.5828 ε hε δ x.left x.right N hN hNpos hN' P x_1.left x_1.right J hJ hx hy this)); Chapter11.integ_of_uniform_cts._proof_4 _fvar.329 _fvar.5679 _fvar.5828 ε hε δ x.left x.right N hN hNpos hN' P x_1.left x_1.right J hJ hx hy this this_1; have hJnon := Eq.mpr (id Set.image_nonempty._simp_1) (Classical.byContradiction fun h => have h := of_eq_true (Eq.trans (Eq.trans (congrArg (fun x => Subsingleton ↑x) (Eq.mp (Mathlib.Tactic.PushNeg.not_nonempty_eq ↑J) h)) (Set.subsingleton_coe._simp_1 ∅)) Set.subsingleton_empty._simp_1); False.elim (Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (Chapter11.BoundedInterval.b _fvar.236)) (Mathlib.Tactic.Ring.atom_pf (Chapter11.BoundedInterval.a _fvar.236)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (Chapter11.BoundedInterval.a _fvar.236) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Chapter11.BoundedInterval.b _fvar.236 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (Chapter11.BoundedInterval.a _fvar.236 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.atom_pf ↑N) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Tactic.Ring.inv_mul (Eq.refl (↑N)⁻¹) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsNNRat.to_isNat (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))))) (Mathlib.Tactic.Ring.mul_pf_right (↑N)⁻¹ (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (Chapter11.BoundedInterval.b _fvar.236) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑N)⁻¹ (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (Chapter11.BoundedInterval.b _fvar.236 ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (Chapter11.BoundedInterval.b _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (Chapter11.BoundedInterval.a _fvar.236) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑N)⁻¹ (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast))) (Mathlib.Tactic.Ring.mul_zero (Chapter11.BoundedInterval.a _fvar.236 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (Chapter11.BoundedInterval.a _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))) (Mathlib.Tactic.Ring.zero_mul ((↑N)⁻¹ ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Chapter11.BoundedInterval.a _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Chapter11.BoundedInterval.b _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (Chapter11.BoundedInterval.a _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0)))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (Chapter11.BoundedInterval.b _fvar.236) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑N)⁻¹ (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (Chapter11.BoundedInterval.a _fvar.236) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑N)⁻¹ (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Int.negOfNat 1).rawCast))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (Chapter11.BoundedInterval.a _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Chapter11.BoundedInterval.b _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (Chapter11.BoundedInterval.a _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0)))) (Mathlib.Tactic.Ring.zero_mul (Chapter11.BoundedInterval.b _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * Nat.rawCast 1) + (Chapter11.BoundedInterval.a _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (Chapter11.BoundedInterval.b _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * Nat.rawCast 1) + (Chapter11.BoundedInterval.a _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.zero_mul 0) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (Chapter11.BoundedInterval.b _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * Nat.rawCast 1) + (Chapter11.BoundedInterval.a _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.zero_mul 0) (Mathlib.Tactic.Ring.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (Chapter11.BoundedInterval.b _fvar.236)) (Mathlib.Tactic.Ring.atom_pf (Chapter11.BoundedInterval.a _fvar.236)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (Chapter11.BoundedInterval.a _fvar.236) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Chapter11.BoundedInterval.b _fvar.236 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (Chapter11.BoundedInterval.a _fvar.236 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.atom_pf ↑N) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Tactic.Ring.inv_mul (Eq.refl (↑N)⁻¹) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsNNRat.to_isNat (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))))) (Mathlib.Tactic.Ring.mul_pf_right (↑N)⁻¹ (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (Chapter11.BoundedInterval.b _fvar.236) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑N)⁻¹ (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (Chapter11.BoundedInterval.b _fvar.236 ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (Chapter11.BoundedInterval.b _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (Chapter11.BoundedInterval.a _fvar.236) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑N)⁻¹ (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast))) (Mathlib.Tactic.Ring.mul_zero (Chapter11.BoundedInterval.a _fvar.236 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (Chapter11.BoundedInterval.a _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))) (Mathlib.Tactic.Ring.zero_mul ((↑N)⁻¹ ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Chapter11.BoundedInterval.a _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Chapter11.BoundedInterval.b _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (Chapter11.BoundedInterval.a _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0)))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (Chapter11.BoundedInterval.b _fvar.236) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑N)⁻¹ (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (Chapter11.BoundedInterval.a _fvar.236) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑N)⁻¹ (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Int.negOfNat 1).rawCast))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (Chapter11.BoundedInterval.a _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Chapter11.BoundedInterval.b _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (Chapter11.BoundedInterval.a _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0)))) (Mathlib.Tactic.Ring.zero_mul (Chapter11.BoundedInterval.b _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * Nat.rawCast 1) + (Chapter11.BoundedInterval.a _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (Chapter11.BoundedInterval.b _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * Nat.rawCast 1) + (Chapter11.BoundedInterval.a _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (Chapter11.BoundedInterval.b _fvar.236) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑N)⁻¹ (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (Chapter11.BoundedInterval.a _fvar.236) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑N)⁻¹ (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (Chapter11.BoundedInterval.b _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (Chapter11.BoundedInterval.a _fvar.236 ^ Nat.rawCast 1 * ((↑N)⁻¹ ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (Chapter11.BoundedInterval.b _fvar.236) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑N)⁻¹ (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (Chapter11.BoundedInterval.a _fvar.236) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑N)⁻¹ (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.lt_of_eq_of_lt (Eq.mp (congrArg (fun _a => _a = 0) (Mathlib.Tactic.Linarith.without_one_mul (CancelDenoms.sub_subst rfl rfl))) (sub_eq_zero_of_eq (Eq.mp (Eq.trans Chapter11.integ_of_uniform_cts._simp_4 (congrArg (fun x => x = 0) (x_1.right J hJ))) h))) (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Linarith.without_one_mul (CancelDenoms.sub_subst rfl rfl))) (Mathlib.Tactic.Linarith.sub_neg_of_lt (have this := div_pos _fvar.5828 (Nat.cast_pos'.mpr hNpos); this))))))); have this := fun y hy => csSup_le hJnon (Chapter11.integ_of_uniform_cts._proof_5 _fvar.329 _fvar.5679 _fvar.5828 ε hε δ x.left x.right N hN hNpos hN' P x_1.left x_1.right J hJ this hJnon y hy); have this := le_csInf hJnon (Chapter11.integ_of_uniform_cts._proof_6 _fvar.329 _fvar.5679 _fvar.5828 ε hε δ x.left x.right N hN hNpos hN' P x_1.left x_1.right J hJ hJnon this); le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (sSup (_fvar.237 '' ↑J))) (Mathlib.Tactic.Ring.atom_pf ε) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul ε (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (sSup (_fvar.237 '' ↑J) ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (ε ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.atom_pf (sInf (_fvar.237 '' ↑J))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (sInf (_fvar.237 '' ↑J)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (sSup (_fvar.237 '' ↑J) ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt (ε ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_zero_add (sInf (_fvar.237 '' ↑J) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ε) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (sSup (_fvar.237 '' ↑J))) (Mathlib.Tactic.Ring.atom_pf (sInf (_fvar.237 '' ↑J))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (sInf (_fvar.237 '' ↑J)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (sSup (_fvar.237 '' ↑J) ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (sInf (_fvar.237 '' ↑J) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (sSup (_fvar.237 '' ↑J)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (sInf (_fvar.237 '' ↑J)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_gt (sSup (_fvar.237 '' ↑J) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_lt (ε ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (sInf (_fvar.237 '' ↑J) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (sSup (_fvar.237 '' ↑J)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero ε (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (sInf (_fvar.237 '' ↑J)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_le_of_neg (Mathlib.Tactic.Linarith.sub_nonpos_of_le this) (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))) (le_max_of_le_right (Mathlib.Meta.Positivity.nonneg_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))) (Chapter11.integ_of_uniform_cts._proof_7 _fvar.329 _fvar.5679 _fvar.5828 ε hε δ x.left x.right N hN hNpos hN' P x_1.left x_1.right)) (Eq.mpr (id (congrArg (fun x => x = ε * (Chapter11.BoundedInterval.b _fvar.236 - Chapter11.BoundedInterval.a _fvar.236)) (Eq.trans (Finset.sum_const (ε * (Chapter11.BoundedInterval.b _fvar.236 - Chapter11.BoundedInterval.a _fvar.236) / ↑N)) (Eq.trans (congrArg (fun x => x • (ε * (Chapter11.BoundedInterval.b _fvar.236 - Chapter11.BoundedInterval.a _fvar.236) / ↑N)) x_1.left) (nsmul_eq_mul N (ε * (Chapter11.BoundedInterval.b _fvar.236 - Chapter11.BoundedInterval.a _fvar.236) / ↑N)))))) (of_eq_true (Eq.trans (congrArg (fun x => x = ε * (Chapter11.BoundedInterval.b _fvar.236 - Chapter11.BoundedInterval.a _fvar.236)) (Eq.trans (mul_div_assoc' (↑N) (ε * (Chapter11.BoundedInterval.b _fvar.236 - Chapter11.BoundedInterval.a _fvar.236)) ↑N) (mul_div_cancel_left₀ (ε * (Chapter11.BoundedInterval.b _fvar.236 - Chapter11.BoundedInterval.a _fvar.236)) (ne_of_gt (Nat.cast_pos'.mpr hNpos))))) (eq_self (ε * (Chapter11.BoundedInterval.b _fvar.236 - Chapter11.BoundedInterval.a _fvar.236))))))) (Classical.choose this) (Classical.choose_spec this)) (Classical.choose (exists_nat_gt ((Chapter11.BoundedInterval.b _fvar.236 - Chapter11.BoundedInterval.a _fvar.236) / δ))) (Classical.choose_spec (exists_nat_gt ((Chapter11.BoundedInterval.b _fvar.236 - Chapter11.BoundedInterval.a _fvar.236) / δ)))) (Classical.choose (Eq.mp (congrArg (fun _a => _a) (propext (Chapter9.UniformContinuousOn.iff _fvar.237 ↑_fvar.236))) _fvar.238 ε hε)) (Classical.choose_spec (Eq.mp (congrArg (fun _a => _a) (propext (Chapter9.UniformContinuousOn.iff _fvar.237 ↑_fvar.236))) _fvar.238 ε hε))lower_le_upper:0 ≤ Chapter11.upper_integral _fvar.237 _fvar.236 - Chapter11.lower_integral _fvar.237 _fvar.236 := le_of_not_gt fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (Chapter11.upper_integral _fvar.237 _fvar.236)) (Mathlib.Tactic.Ring.atom_pf (Chapter11.lower_integral _fvar.237 _fvar.236)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (Chapter11.lower_integral _fvar.237 _fvar.236) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Chapter11.upper_integral _fvar.237 _fvar.236 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (Chapter11.lower_integral _fvar.237 _fvar.236 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (Chapter11.upper_integral _fvar.237 _fvar.236 ^ Nat.rawCast 1 * Nat.rawCast 1 + (Chapter11.lower_integral _fvar.237 _fvar.236 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (Chapter11.lower_integral _fvar.237 _fvar.236)) (Mathlib.Tactic.Ring.atom_pf (Chapter11.upper_integral _fvar.237 _fvar.236)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (Chapter11.upper_integral _fvar.237 _fvar.236) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Chapter11.upper_integral _fvar.237 _fvar.236 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_zero (Chapter11.lower_integral _fvar.237 _fvar.236 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (Chapter11.upper_integral _fvar.237 _fvar.236) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (Chapter11.lower_integral _fvar.237 _fvar.236) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt a) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Chapter11.lower_integral_le_upper _fvar.329))))h:0 < upper_integral f I - lower_integral f Iε:ℝ := (Chapter11.upper_integral _fvar.237 _fvar.236 - Chapter11.lower_integral _fvar.237 _fvar.236) / (2 * (_fvar.5638 - _fvar.5555))⊢ ε > 0 All goals completed! 🐙) using 1; All goals completed! 🐙 All goals completed! 🐙 All goals completed! 🐙

Corollary 11.5.2

theorem integ_of_cts {a b:ℝ} {f:ℝ → ℝ} (hf: ContinuousOn f (Icc a b)) : IntegrableOn f (Icc a b) := integ_of_uniform_cts (UniformContinuousOn.of_continuousOn hf)
declaration uses 'sorry'example : ContinuousOn (fun x:ℝ ↦ 1/x) (Icc 0 1) := ⊢ ContinuousOn (fun x => 1 / x) ↑(Icc 0 1) All goals completed! 🐙declaration uses 'sorry'example : ¬ IntegrableOn (fun x:ℝ ↦ 1/x) (Icc 0 1) := ⊢ ¬IntegrableOn (fun x => 1 / x) (Icc 0 1) All goals completed! 🐙open PiecewiseConstantOn ConstantOn in set_option maxHeartbeats 300000 in /-- Proposition 11.5.3-/ theorem integ_of_bdd_cts {I: BoundedInterval} {f:ℝ → ℝ} (hbound: BddOn f I) (hf: ContinuousOn f I) : IntegrableOn f I := I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑I⊢ IntegrableOn f I -- This proof is written to follow the structure of the original text. I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑Ihsing:I.length = 0⊢ IntegrableOn f II:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑Ihsing:¬I.length = 0⊢ IntegrableOn f I I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑Ihsing:I.length = 0⊢ IntegrableOn f I All goals completed! 🐙 have hI : (I:Set ℝ).Nonempty := I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑I⊢ IntegrableOn f I I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑Ihsing:¬I.length = 0this:↑I = ∅⊢ False; I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑Ihsing:¬Subsingleton ↑↑Ithis:↑I = ∅⊢ False; All goals completed! 🐙 I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796hsing:I.a < I.b⊢ IntegrableOn f I I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611hsing:a < I.b⊢ IntegrableOn f I I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < b⊢ IntegrableOn f I I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613⊢ IntegrableOn f I I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ M⊢ IntegrableOn f I I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ _fvar.207620 := LE.le.trans (abs_nonneg ?_mvar.207803) (@_fvar.207621 (Set.Nonempty.some _fvar.200797) (Set.Nonempty.some_mem _fvar.200797))⊢ IntegrableOn f I have (ε:ℝ) (hε: ε > 0) : upper_integral f I - lower_integral f I ≤ (4*M+2) * ε := I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑I⊢ IntegrableOn f I I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ _fvar.207620 := LE.le.trans (abs_nonneg ?_mvar.207803) (@_fvar.207621 (Set.Nonempty.some _fvar.200797) (Set.Nonempty.some_mem _fvar.200797))ε:ℝhε:ε > 0this:∀ {I : BoundedInterval} {f : ℝ → ℝ}, BddOn f ↑I → ContinuousOn f ↑I → (↑I).Nonempty → let a := I.a; let b := I.b; a < b → lower_integral f I ≤ upper_integral f I → ∀ (M : ℝ), (∀ x ∈ ↑I, |f x| ≤ M) → 0 ≤ M → ∀ ε > 0, ε < (b - a) / 2 → upper_integral f I - lower_integral f I ≤ (4 * M + 2) * εhε':¬ε < (b - a) / 2⊢ upper_integral f I - lower_integral f I ≤ (4 * M + 2) * εI:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑I).Nonemptya:ℝ := Chapter11.BoundedInterval.a _fvar.208937b:ℝ := Chapter11.BoundedInterval.b _fvar.208937hsing:a < blower_le_upper:lower_integral f I ≤ upper_integral f IM:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ Mε:ℝhε:ε > 0hε':ε < (b - a) / 2⊢ upper_integral f I - lower_integral f I ≤ (4 * M + 2) * ε I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ _fvar.207620 := LE.le.trans (abs_nonneg ?_mvar.207803) (@_fvar.207621 (Set.Nonempty.some _fvar.200797) (Set.Nonempty.some_mem _fvar.200797))ε:ℝhε:ε > 0this:∀ {I : BoundedInterval} {f : ℝ → ℝ}, BddOn f ↑I → ContinuousOn f ↑I → (↑I).Nonempty → let a := I.a; let b := I.b; a < b → lower_integral f I ≤ upper_integral f I → ∀ (M : ℝ), (∀ x ∈ ↑I, |f x| ≤ M) → 0 ≤ M → ∀ ε > 0, ε < (b - a) / 2 → upper_integral f I - lower_integral f I ≤ (4 * M + 2) * εhε':¬ε < (b - a) / 2⊢ upper_integral f I - lower_integral f I ≤ (4 * M + 2) * ε I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ _fvar.207620 := LE.le.trans (abs_nonneg ?_mvar.207803) (@_fvar.207621 (Set.Nonempty.some _fvar.200797) (Set.Nonempty.some_mem _fvar.200797))ε:ℝhε:ε > 0this:∀ {I : BoundedInterval} {f : ℝ → ℝ}, BddOn f ↑I → ContinuousOn f ↑I → (↑I).Nonempty → let a := I.a; let b := I.b; a < b → lower_integral f I ≤ upper_integral f I → ∀ (M : ℝ), (∀ x ∈ ↑I, |f x| ≤ M) → 0 ≤ M → ∀ ε > 0, ε < (b - a) / 2 → upper_integral f I - lower_integral f I ≤ (4 * M + 2) * εhε':¬ε < (b - a) / 2⊢ BddOn f ↑II:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ _fvar.207620 := LE.le.trans (abs_nonneg ?_mvar.207803) (@_fvar.207621 (Set.Nonempty.some _fvar.200797) (Set.Nonempty.some_mem _fvar.200797))ε:ℝhε:ε > 0this:∀ {I : BoundedInterval} {f : ℝ → ℝ}, BddOn f ↑I → ContinuousOn f ↑I → (↑I).Nonempty → let a := I.a; let b := I.b; a < b → lower_integral f I ≤ upper_integral f I → ∀ (M : ℝ), (∀ x ∈ ↑I, |f x| ≤ M) → 0 ≤ M → ∀ ε > 0, ε < (b - a) / 2 → upper_integral f I - lower_integral f I ≤ (4 * M + 2) * εhε':¬ε < (b - a) / 2⊢ ContinuousOn f ↑II:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ _fvar.207620 := LE.le.trans (abs_nonneg ?_mvar.207803) (@_fvar.207621 (Set.Nonempty.some _fvar.200797) (Set.Nonempty.some_mem _fvar.200797))ε:ℝhε:ε > 0this:∀ {I : BoundedInterval} {f : ℝ → ℝ}, BddOn f ↑I → ContinuousOn f ↑I → (↑I).Nonempty → let a := I.a; let b := I.b; a < b → lower_integral f I ≤ upper_integral f I → ∀ (M : ℝ), (∀ x ∈ ↑I, |f x| ≤ M) → 0 ≤ M → ∀ ε > 0, ε < (b - a) / 2 → upper_integral f I - lower_integral f I ≤ (4 * M + 2) * εhε':¬ε < (b - a) / 2⊢ (↑I).NonemptyI:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ _fvar.207620 := LE.le.trans (abs_nonneg ?_mvar.207803) (@_fvar.207621 (Set.Nonempty.some _fvar.200797) (Set.Nonempty.some_mem _fvar.200797))ε:ℝhε:ε > 0this:∀ {I : BoundedInterval} {f : ℝ → ℝ}, BddOn f ↑I → ContinuousOn f ↑I → (↑I).Nonempty → let a := I.a; let b := I.b; a < b → lower_integral f I ≤ upper_integral f I → ∀ (M : ℝ), (∀ x ∈ ↑I, |f x| ≤ M) → 0 ≤ M → ∀ ε > 0, ε < (b - a) / 2 → upper_integral f I - lower_integral f I ≤ (4 * M + 2) * εhε':¬ε < (b - a) / 2⊢ I.a < I.bI:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ _fvar.207620 := LE.le.trans (abs_nonneg ?_mvar.207803) (@_fvar.207621 (Set.Nonempty.some _fvar.200797) (Set.Nonempty.some_mem _fvar.200797))ε:ℝhε:ε > 0this:∀ {I : BoundedInterval} {f : ℝ → ℝ}, BddOn f ↑I → ContinuousOn f ↑I → (↑I).Nonempty → let a := I.a; let b := I.b; a < b → lower_integral f I ≤ upper_integral f I → ∀ (M : ℝ), (∀ x ∈ ↑I, |f x| ≤ M) → 0 ≤ M → ∀ ε > 0, ε < (b - a) / 2 → upper_integral f I - lower_integral f I ≤ (4 * M + 2) * εhε':¬ε < (b - a) / 2⊢ lower_integral f I ≤ upper_integral f II:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ _fvar.207620 := LE.le.trans (abs_nonneg ?_mvar.207803) (@_fvar.207621 (Set.Nonempty.some _fvar.200797) (Set.Nonempty.some_mem _fvar.200797))ε:ℝhε:ε > 0this:∀ {I : BoundedInterval} {f : ℝ → ℝ}, BddOn f ↑I → ContinuousOn f ↑I → (↑I).Nonempty → let a := I.a; let b := I.b; a < b → lower_integral f I ≤ upper_integral f I → ∀ (M : ℝ), (∀ x ∈ ↑I, |f x| ≤ M) → 0 ≤ M → ∀ ε > 0, ε < (b - a) / 2 → upper_integral f I - lower_integral f I ≤ (4 * M + 2) * εhε':¬ε < (b - a) / 2⊢ 0 ≤ MI:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ _fvar.207620 := LE.le.trans (abs_nonneg ?_mvar.207803) (@_fvar.207621 (Set.Nonempty.some _fvar.200797) (Set.Nonempty.some_mem _fvar.200797))ε:ℝhε:ε > 0this:∀ {I : BoundedInterval} {f : ℝ → ℝ}, BddOn f ↑I → ContinuousOn f ↑I → (↑I).Nonempty → let a := I.a; let b := I.b; a < b → lower_integral f I ≤ upper_integral f I → ∀ (M : ℝ), (∀ x ∈ ↑I, |f x| ≤ M) → 0 ≤ M → ∀ ε > 0, ε < (b - a) / 2 → upper_integral f I - lower_integral f I ≤ (4 * M + 2) * εhε':¬ε < (b - a) / 2⊢ (b - a) / 3 > 0I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ _fvar.207620 := LE.le.trans (abs_nonneg ?_mvar.207803) (@_fvar.207621 (Set.Nonempty.some _fvar.200797) (Set.Nonempty.some_mem _fvar.200797))ε:ℝhε:ε > 0this:∀ {I : BoundedInterval} {f : ℝ → ℝ}, BddOn f ↑I → ContinuousOn f ↑I → (↑I).Nonempty → let a := I.a; let b := I.b; a < b → lower_integral f I ≤ upper_integral f I → ∀ (M : ℝ), (∀ x ∈ ↑I, |f x| ≤ M) → 0 ≤ M → ∀ ε > 0, ε < (b - a) / 2 → upper_integral f I - lower_integral f I ≤ (4 * M + 2) * εhε':¬ε < (b - a) / 2⊢ (b - a) / 3 < (I.b - I.a) / 2I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ _fvar.207620 := LE.le.trans (abs_nonneg ?_mvar.207803) (@_fvar.207621 (Set.Nonempty.some _fvar.200797) (Set.Nonempty.some_mem _fvar.200797))ε:ℝhε:ε > 0hε':¬ε < (b - a) / 2this:upper_integral f I - lower_integral f I ≤ (4 * M + 2) * ((b - a) / 3)⊢ upper_integral f I - lower_integral f I ≤ (4 * M + 2) * ε I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ _fvar.207620 := LE.le.trans (abs_nonneg ?_mvar.207803) (@_fvar.207621 (Set.Nonempty.some _fvar.200797) (Set.Nonempty.some_mem _fvar.200797))ε:ℝhε:ε > 0this:∀ {I : BoundedInterval} {f : ℝ → ℝ}, BddOn f ↑I → ContinuousOn f ↑I → (↑I).Nonempty → let a := I.a; let b := I.b; a < b → lower_integral f I ≤ upper_integral f I → ∀ (M : ℝ), (∀ x ∈ ↑I, |f x| ≤ M) → 0 ≤ M → ∀ ε > 0, ε < (b - a) / 2 → upper_integral f I - lower_integral f I ≤ (4 * M + 2) * εhε':¬ε < (b - a) / 2⊢ BddOn f ↑II:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ _fvar.207620 := LE.le.trans (abs_nonneg ?_mvar.207803) (@_fvar.207621 (Set.Nonempty.some _fvar.200797) (Set.Nonempty.some_mem _fvar.200797))ε:ℝhε:ε > 0this:∀ {I : BoundedInterval} {f : ℝ → ℝ}, BddOn f ↑I → ContinuousOn f ↑I → (↑I).Nonempty → let a := I.a; let b := I.b; a < b → lower_integral f I ≤ upper_integral f I → ∀ (M : ℝ), (∀ x ∈ ↑I, |f x| ≤ M) → 0 ≤ M → ∀ ε > 0, ε < (b - a) / 2 → upper_integral f I - lower_integral f I ≤ (4 * M + 2) * εhε':¬ε < (b - a) / 2⊢ ContinuousOn f ↑II:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ _fvar.207620 := LE.le.trans (abs_nonneg ?_mvar.207803) (@_fvar.207621 (Set.Nonempty.some _fvar.200797) (Set.Nonempty.some_mem _fvar.200797))ε:ℝhε:ε > 0this:∀ {I : BoundedInterval} {f : ℝ → ℝ}, BddOn f ↑I → ContinuousOn f ↑I → (↑I).Nonempty → let a := I.a; let b := I.b; a < b → lower_integral f I ≤ upper_integral f I → ∀ (M : ℝ), (∀ x ∈ ↑I, |f x| ≤ M) → 0 ≤ M → ∀ ε > 0, ε < (b - a) / 2 → upper_integral f I - lower_integral f I ≤ (4 * M + 2) * εhε':¬ε < (b - a) / 2⊢ (↑I).NonemptyI:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ _fvar.207620 := LE.le.trans (abs_nonneg ?_mvar.207803) (@_fvar.207621 (Set.Nonempty.some _fvar.200797) (Set.Nonempty.some_mem _fvar.200797))ε:ℝhε:ε > 0this:∀ {I : BoundedInterval} {f : ℝ → ℝ}, BddOn f ↑I → ContinuousOn f ↑I → (↑I).Nonempty → let a := I.a; let b := I.b; a < b → lower_integral f I ≤ upper_integral f I → ∀ (M : ℝ), (∀ x ∈ ↑I, |f x| ≤ M) → 0 ≤ M → ∀ ε > 0, ε < (b - a) / 2 → upper_integral f I - lower_integral f I ≤ (4 * M + 2) * εhε':¬ε < (b - a) / 2⊢ I.a < I.bI:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ _fvar.207620 := LE.le.trans (abs_nonneg ?_mvar.207803) (@_fvar.207621 (Set.Nonempty.some _fvar.200797) (Set.Nonempty.some_mem _fvar.200797))ε:ℝhε:ε > 0this:∀ {I : BoundedInterval} {f : ℝ → ℝ}, BddOn f ↑I → ContinuousOn f ↑I → (↑I).Nonempty → let a := I.a; let b := I.b; a < b → lower_integral f I ≤ upper_integral f I → ∀ (M : ℝ), (∀ x ∈ ↑I, |f x| ≤ M) → 0 ≤ M → ∀ ε > 0, ε < (b - a) / 2 → upper_integral f I - lower_integral f I ≤ (4 * M + 2) * εhε':¬ε < (b - a) / 2⊢ lower_integral f I ≤ upper_integral f II:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ _fvar.207620 := LE.le.trans (abs_nonneg ?_mvar.207803) (@_fvar.207621 (Set.Nonempty.some _fvar.200797) (Set.Nonempty.some_mem _fvar.200797))ε:ℝhε:ε > 0this:∀ {I : BoundedInterval} {f : ℝ → ℝ}, BddOn f ↑I → ContinuousOn f ↑I → (↑I).Nonempty → let a := I.a; let b := I.b; a < b → lower_integral f I ≤ upper_integral f I → ∀ (M : ℝ), (∀ x ∈ ↑I, |f x| ≤ M) → 0 ≤ M → ∀ ε > 0, ε < (b - a) / 2 → upper_integral f I - lower_integral f I ≤ (4 * M + 2) * εhε':¬ε < (b - a) / 2⊢ 0 ≤ MI:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ _fvar.207620 := LE.le.trans (abs_nonneg ?_mvar.207803) (@_fvar.207621 (Set.Nonempty.some _fvar.200797) (Set.Nonempty.some_mem _fvar.200797))ε:ℝhε:ε > 0this:∀ {I : BoundedInterval} {f : ℝ → ℝ}, BddOn f ↑I → ContinuousOn f ↑I → (↑I).Nonempty → let a := I.a; let b := I.b; a < b → lower_integral f I ≤ upper_integral f I → ∀ (M : ℝ), (∀ x ∈ ↑I, |f x| ≤ M) → 0 ≤ M → ∀ ε > 0, ε < (b - a) / 2 → upper_integral f I - lower_integral f I ≤ (4 * M + 2) * εhε':¬ε < (b - a) / 2⊢ (b - a) / 3 > 0I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ _fvar.207620 := LE.le.trans (abs_nonneg ?_mvar.207803) (@_fvar.207621 (Set.Nonempty.some _fvar.200797) (Set.Nonempty.some_mem _fvar.200797))ε:ℝhε:ε > 0this:∀ {I : BoundedInterval} {f : ℝ → ℝ}, BddOn f ↑I → ContinuousOn f ↑I → (↑I).Nonempty → let a := I.a; let b := I.b; a < b → lower_integral f I ≤ upper_integral f I → ∀ (M : ℝ), (∀ x ∈ ↑I, |f x| ≤ M) → 0 ≤ M → ∀ ε > 0, ε < (b - a) / 2 → upper_integral f I - lower_integral f I ≤ (4 * M + 2) * εhε':¬ε < (b - a) / 2⊢ (b - a) / 3 < (I.b - I.a) / 2I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ _fvar.207620 := LE.le.trans (abs_nonneg ?_mvar.207803) (@_fvar.207621 (Set.Nonempty.some _fvar.200797) (Set.Nonempty.some_mem _fvar.200797))ε:ℝhε:ε > 0hε':¬ε < (b - a) / 2this:upper_integral f I - lower_integral f I ≤ (4 * M + 2) * ((b - a) / 3)⊢ upper_integral f I - lower_integral f I ≤ (4 * M + 2) * ε first | I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ _fvar.207620 := LE.le.trans (abs_nonneg ?_mvar.207803) (@_fvar.207621 (Set.Nonempty.some _fvar.200797) (Set.Nonempty.some_mem _fvar.200797))ε:ℝhε:ε > 0hε':¬ε < (b - a) / 2this:upper_integral f I - lower_integral f I ≤ (4 * M + 2) * ((b - a) / 3)⊢ upper_integral f I - lower_integral f I ≤ (4 * M + 2) * ε | I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ _fvar.207620 := LE.le.trans (abs_nonneg ?_mvar.207803) (@_fvar.207621 (Set.Nonempty.some _fvar.200797) (Set.Nonempty.some_mem _fvar.200797))ε:ℝhε:ε > 0hε':¬ε < (b - a) / 2this:upper_integral f I - lower_integral f I ≤ (4 * M + 2) * ((b - a) / 3)⊢ upper_integral f I - lower_integral f I ≤ (4 * M + 2) * ε | I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ _fvar.207620 := LE.le.trans (abs_nonneg ?_mvar.207803) (@_fvar.207621 (Set.Nonempty.some _fvar.200797) (Set.Nonempty.some_mem _fvar.200797))ε:ℝhε:ε > 0hε':¬ε < (b - a) / 2this:upper_integral f I - lower_integral f I ≤ (4 * M + 2) * ((b - a) / 3)⊢ (4 * M + 2) * ((b - a) / 3) ≤ (4 * M + 2) * ε; I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑_fvar.200611).Nonempty := ?_mvar.200796a:ℝ := Chapter11.BoundedInterval.a _fvar.200611b:ℝ := Chapter11.BoundedInterval.b _fvar.200611hsing:a < blower_le_upper:?_mvar.207565 := Chapter11.lower_integral_le_upper _fvar.200613M:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ _fvar.207620 := LE.le.trans (abs_nonneg ?_mvar.207803) (@_fvar.207621 (Set.Nonempty.some _fvar.200797) (Set.Nonempty.some_mem _fvar.200797))ε:ℝhε:ε > 0hε':¬ε < (b - a) / 2this:upper_integral f I - lower_integral f I ≤ (4 * M + 2) * ((b - a) / 3)⊢ (b - a) / 3 ≤ ε; All goals completed! 🐙 I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑I).Nonemptya:ℝ := Chapter11.BoundedInterval.a _fvar.208937b:ℝ := Chapter11.BoundedInterval.b _fvar.208937hsing:a < blower_le_upper:lower_integral f I ≤ upper_integral f IM:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ Mε:ℝhε:ε > 0hε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.208942 + _fvar.208949) (_fvar.208943 - _fvar.208949)⊢ upper_integral f I - lower_integral f I ≤ (4 * M + 2) * ε I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑I).Nonemptya:ℝ := Chapter11.BoundedInterval.a _fvar.208937b:ℝ := Chapter11.BoundedInterval.b _fvar.208937hsing:a < blower_le_upper:lower_integral f I ≤ upper_integral f IM:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ Mε:ℝhε:ε > 0hε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.208942 + _fvar.208949) (_fvar.208943 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match _fvar.208937, _fvar.208939, _fvar.208940, _fvar.208941, _fvar.208945, _fvar.208947 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.208942 (_fvar.208942 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.208942 (_fvar.208942 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.208942 (_fvar.208942 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.208942 (_fvar.208942 + _fvar.208949)⊢ upper_integral f I - lower_integral f I ≤ (4 * M + 2) * ε I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑I).Nonemptya:ℝ := Chapter11.BoundedInterval.a _fvar.208937b:ℝ := Chapter11.BoundedInterval.b _fvar.208937hsing:a < blower_le_upper:lower_integral f I ≤ upper_integral f IM:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ Mε:ℝhε:ε > 0hε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.208942 + _fvar.208949) (_fvar.208943 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match _fvar.208937, _fvar.208939, _fvar.208940, _fvar.208941, _fvar.208945, _fvar.208947 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.208942 (_fvar.208942 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.208942 (_fvar.208942 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.208942 (_fvar.208942 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.208942 (_fvar.208942 + _fvar.208949)Iright:Chapter11.BoundedInterval := match _fvar.208937, _fvar.208939, _fvar.208940, _fvar.208941, _fvar.208945, _fvar.208947 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.208943 - _fvar.208949) _fvar.208943 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.208943 - _fvar.208949) _fvar.208943 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.208943 - _fvar.208949) _fvar.208943 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.208943 - _fvar.208949) _fvar.208943⊢ upper_integral f I - lower_integral f I ≤ (4 * M + 2) * ε I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑IhI:(↑I).Nonemptya:ℝ := Chapter11.BoundedInterval.a _fvar.208937b:ℝ := Chapter11.BoundedInterval.b _fvar.208937hsing:a < blower_le_upper:lower_integral f I ≤ upper_integral f IM:ℝhM:∀ x ∈ ↑I, |f x| ≤ MhMpos:0 ≤ Mε:ℝhε:ε > 0hε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.208942 + _fvar.208949) (_fvar.208943 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match _fvar.208937, _fvar.208939, _fvar.208940, _fvar.208941, _fvar.208945, _fvar.208947 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.208942 (_fvar.208942 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.208942 (_fvar.208942 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.208942 (_fvar.208942 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.208942 (_fvar.208942 + _fvar.208949)Iright:Chapter11.BoundedInterval := match _fvar.208937, _fvar.208939, _fvar.208940, _fvar.208941, _fvar.208945, _fvar.208947 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.208943 - _fvar.208949) _fvar.208943 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.208943 - _fvar.208949) _fvar.208943 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.208943 - _fvar.208949) _fvar.208943 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.208943 - _fvar.208949) _fvar.208943Ileft':Chapter11.BoundedInterval := match _fvar.208937, _fvar.208939, _fvar.208940, _fvar.208941, _fvar.208945, _fvar.208947 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.208942 (_fvar.208943 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.208942 (_fvar.208943 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.208942 (_fvar.208943 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.208942 (_fvar.208943 - _fvar.208949)⊢ upper_integral f I - lower_integral f I ≤ (4 * M + 2) * ε have Ileftlen : |Ileft|ₗ = ε := I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑I⊢ IntegrableOn f I f:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ioo a✝ b✝)hf:ContinuousOn f ↑(Ioo a✝ b✝)hI:(↑(Ioo a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ioo _fvar.233448 _fvar.233449).ab:ℝ := (Chapter11.BoundedInterval.Ioo _fvar.233448 _fvar.233449).bhsing:a < blower_le_upper:lower_integral f (Ioo a✝ b✝) ≤ upper_integral f (Ioo a✝ b✝)hM:∀ x ∈ ↑(Ioo a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.233484 + _fvar.208949) (_fvar.233485 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.233448 _fvar.233449, _fvar.233481, _fvar.233482, _fvar.233483, _fvar.233487, _fvar.233488 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.233484 (_fvar.233484 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.233484 (_fvar.233484 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.233484 (_fvar.233484 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.233484 (_fvar.233484 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.233448 _fvar.233449, _fvar.233481, _fvar.233482, _fvar.233483, _fvar.233487, _fvar.233488 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.233485 - _fvar.208949) _fvar.233485 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.233485 - _fvar.208949) _fvar.233485 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.233485 - _fvar.208949) _fvar.233485 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.233485 - _fvar.208949) _fvar.233485Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.233448 _fvar.233449, _fvar.233481, _fvar.233482, _fvar.233483, _fvar.233487, _fvar.233488 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.233484 (_fvar.233485 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.233484 (_fvar.233485 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.233484 (_fvar.233485 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.233484 (_fvar.233485 - _fvar.208949)⊢ Ileft.length = εf:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Icc a✝ b✝)hf:ContinuousOn f ↑(Icc a✝ b✝)hI:(↑(Icc a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Icc _fvar.233498 _fvar.233499).ab:ℝ := (Chapter11.BoundedInterval.Icc _fvar.233498 _fvar.233499).bhsing:a < blower_le_upper:lower_integral f (Icc a✝ b✝) ≤ upper_integral f (Icc a✝ b✝)hM:∀ x ∈ ↑(Icc a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.233534 + _fvar.208949) (_fvar.233535 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.233498 _fvar.233499, _fvar.233531, _fvar.233532, _fvar.233533, _fvar.233537, _fvar.233538 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.233534 (_fvar.233534 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.233534 (_fvar.233534 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.233534 (_fvar.233534 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.233534 (_fvar.233534 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.233498 _fvar.233499, _fvar.233531, _fvar.233532, _fvar.233533, _fvar.233537, _fvar.233538 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.233535 - _fvar.208949) _fvar.233535 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.233535 - _fvar.208949) _fvar.233535 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.233535 - _fvar.208949) _fvar.233535 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.233535 - _fvar.208949) _fvar.233535Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.233498 _fvar.233499, _fvar.233531, _fvar.233532, _fvar.233533, _fvar.233537, _fvar.233538 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.233534 (_fvar.233535 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.233534 (_fvar.233535 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.233534 (_fvar.233535 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.233534 (_fvar.233535 - _fvar.208949)⊢ Ileft.length = εf:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ioc a✝ b✝)hf:ContinuousOn f ↑(Ioc a✝ b✝)hI:(↑(Ioc a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ioc _fvar.233548 _fvar.233549).ab:ℝ := (Chapter11.BoundedInterval.Ioc _fvar.233548 _fvar.233549).bhsing:a < blower_le_upper:lower_integral f (Ioc a✝ b✝) ≤ upper_integral f (Ioc a✝ b✝)hM:∀ x ∈ ↑(Ioc a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.233584 + _fvar.208949) (_fvar.233585 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.233548 _fvar.233549, _fvar.233581, _fvar.233582, _fvar.233583, _fvar.233587, _fvar.233588 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.233584 (_fvar.233584 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.233584 (_fvar.233584 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.233584 (_fvar.233584 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.233584 (_fvar.233584 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.233548 _fvar.233549, _fvar.233581, _fvar.233582, _fvar.233583, _fvar.233587, _fvar.233588 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.233585 - _fvar.208949) _fvar.233585 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.233585 - _fvar.208949) _fvar.233585 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.233585 - _fvar.208949) _fvar.233585 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.233585 - _fvar.208949) _fvar.233585Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.233548 _fvar.233549, _fvar.233581, _fvar.233582, _fvar.233583, _fvar.233587, _fvar.233588 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.233584 (_fvar.233585 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.233584 (_fvar.233585 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.233584 (_fvar.233585 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.233584 (_fvar.233585 - _fvar.208949)⊢ Ileft.length = εf:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ico a✝ b✝)hf:ContinuousOn f ↑(Ico a✝ b✝)hI:(↑(Ico a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ico _fvar.233598 _fvar.233599).ab:ℝ := (Chapter11.BoundedInterval.Ico _fvar.233598 _fvar.233599).bhsing:a < blower_le_upper:lower_integral f (Ico a✝ b✝) ≤ upper_integral f (Ico a✝ b✝)hM:∀ x ∈ ↑(Ico a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.233634 + _fvar.208949) (_fvar.233635 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.233598 _fvar.233599, _fvar.233631, _fvar.233632, _fvar.233633, _fvar.233637, _fvar.233638 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.233634 (_fvar.233634 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.233634 (_fvar.233634 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.233634 (_fvar.233634 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.233634 (_fvar.233634 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.233598 _fvar.233599, _fvar.233631, _fvar.233632, _fvar.233633, _fvar.233637, _fvar.233638 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.233635 - _fvar.208949) _fvar.233635 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.233635 - _fvar.208949) _fvar.233635 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.233635 - _fvar.208949) _fvar.233635 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.233635 - _fvar.208949) _fvar.233635Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.233598 _fvar.233599, _fvar.233631, _fvar.233632, _fvar.233633, _fvar.233637, _fvar.233638 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.233634 (_fvar.233635 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.233634 (_fvar.233635 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.233634 (_fvar.233635 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.233634 (_fvar.233635 - _fvar.208949)⊢ Ileft.length = ε f:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ioo a✝ b✝)hf:ContinuousOn f ↑(Ioo a✝ b✝)hI:(↑(Ioo a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ioo _fvar.233448 _fvar.233449).ab:ℝ := (Chapter11.BoundedInterval.Ioo _fvar.233448 _fvar.233449).bhsing:a < blower_le_upper:lower_integral f (Ioo a✝ b✝) ≤ upper_integral f (Ioo a✝ b✝)hM:∀ x ∈ ↑(Ioo a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.233484 + _fvar.208949) (_fvar.233485 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.233448 _fvar.233449, _fvar.233481, _fvar.233482, _fvar.233483, _fvar.233487, _fvar.233488 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.233484 (_fvar.233484 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.233484 (_fvar.233484 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.233484 (_fvar.233484 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.233484 (_fvar.233484 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.233448 _fvar.233449, _fvar.233481, _fvar.233482, _fvar.233483, _fvar.233487, _fvar.233488 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.233485 - _fvar.208949) _fvar.233485 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.233485 - _fvar.208949) _fvar.233485 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.233485 - _fvar.208949) _fvar.233485 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.233485 - _fvar.208949) _fvar.233485Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.233448 _fvar.233449, _fvar.233481, _fvar.233482, _fvar.233483, _fvar.233487, _fvar.233488 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.233484 (_fvar.233485 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.233484 (_fvar.233485 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.233484 (_fvar.233485 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.233484 (_fvar.233485 - _fvar.208949)⊢ Ileft.length = εf:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Icc a✝ b✝)hf:ContinuousOn f ↑(Icc a✝ b✝)hI:(↑(Icc a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Icc _fvar.233498 _fvar.233499).ab:ℝ := (Chapter11.BoundedInterval.Icc _fvar.233498 _fvar.233499).bhsing:a < blower_le_upper:lower_integral f (Icc a✝ b✝) ≤ upper_integral f (Icc a✝ b✝)hM:∀ x ∈ ↑(Icc a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.233534 + _fvar.208949) (_fvar.233535 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.233498 _fvar.233499, _fvar.233531, _fvar.233532, _fvar.233533, _fvar.233537, _fvar.233538 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.233534 (_fvar.233534 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.233534 (_fvar.233534 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.233534 (_fvar.233534 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.233534 (_fvar.233534 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.233498 _fvar.233499, _fvar.233531, _fvar.233532, _fvar.233533, _fvar.233537, _fvar.233538 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.233535 - _fvar.208949) _fvar.233535 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.233535 - _fvar.208949) _fvar.233535 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.233535 - _fvar.208949) _fvar.233535 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.233535 - _fvar.208949) _fvar.233535Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.233498 _fvar.233499, _fvar.233531, _fvar.233532, _fvar.233533, _fvar.233537, _fvar.233538 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.233534 (_fvar.233535 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.233534 (_fvar.233535 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.233534 (_fvar.233535 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.233534 (_fvar.233535 - _fvar.208949)⊢ Ileft.length = εf:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ioc a✝ b✝)hf:ContinuousOn f ↑(Ioc a✝ b✝)hI:(↑(Ioc a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ioc _fvar.233548 _fvar.233549).ab:ℝ := (Chapter11.BoundedInterval.Ioc _fvar.233548 _fvar.233549).bhsing:a < blower_le_upper:lower_integral f (Ioc a✝ b✝) ≤ upper_integral f (Ioc a✝ b✝)hM:∀ x ∈ ↑(Ioc a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.233584 + _fvar.208949) (_fvar.233585 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.233548 _fvar.233549, _fvar.233581, _fvar.233582, _fvar.233583, _fvar.233587, _fvar.233588 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.233584 (_fvar.233584 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.233584 (_fvar.233584 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.233584 (_fvar.233584 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.233584 (_fvar.233584 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.233548 _fvar.233549, _fvar.233581, _fvar.233582, _fvar.233583, _fvar.233587, _fvar.233588 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.233585 - _fvar.208949) _fvar.233585 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.233585 - _fvar.208949) _fvar.233585 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.233585 - _fvar.208949) _fvar.233585 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.233585 - _fvar.208949) _fvar.233585Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.233548 _fvar.233549, _fvar.233581, _fvar.233582, _fvar.233583, _fvar.233587, _fvar.233588 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.233584 (_fvar.233585 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.233584 (_fvar.233585 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.233584 (_fvar.233585 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.233584 (_fvar.233585 - _fvar.208949)⊢ Ileft.length = εf:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ico a✝ b✝)hf:ContinuousOn f ↑(Ico a✝ b✝)hI:(↑(Ico a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ico _fvar.233598 _fvar.233599).ab:ℝ := (Chapter11.BoundedInterval.Ico _fvar.233598 _fvar.233599).bhsing:a < blower_le_upper:lower_integral f (Ico a✝ b✝) ≤ upper_integral f (Ico a✝ b✝)hM:∀ x ∈ ↑(Ico a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.233634 + _fvar.208949) (_fvar.233635 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.233598 _fvar.233599, _fvar.233631, _fvar.233632, _fvar.233633, _fvar.233637, _fvar.233638 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.233634 (_fvar.233634 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.233634 (_fvar.233634 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.233634 (_fvar.233634 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.233634 (_fvar.233634 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.233598 _fvar.233599, _fvar.233631, _fvar.233632, _fvar.233633, _fvar.233637, _fvar.233638 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.233635 - _fvar.208949) _fvar.233635 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.233635 - _fvar.208949) _fvar.233635 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.233635 - _fvar.208949) _fvar.233635 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.233635 - _fvar.208949) _fvar.233635Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.233598 _fvar.233599, _fvar.233631, _fvar.233632, _fvar.233633, _fvar.233637, _fvar.233638 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.233634 (_fvar.233635 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.233634 (_fvar.233635 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.233634 (_fvar.233635 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.233634 (_fvar.233635 - _fvar.208949)⊢ Ileft.length = ε All goals completed! 🐙 have Irightlen : |Iright|ₗ = ε := I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑I⊢ IntegrableOn f I f:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ioo a✝ b✝)hf:ContinuousOn f ↑(Ioo a✝ b✝)hI:(↑(Ioo a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ioo _fvar.237451 _fvar.237452).ab:ℝ := (Chapter11.BoundedInterval.Ioo _fvar.237451 _fvar.237452).bhsing:a < blower_le_upper:lower_integral f (Ioo a✝ b✝) ≤ upper_integral f (Ioo a✝ b✝)hM:∀ x ∈ ↑(Ioo a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.237488 + _fvar.208949) (_fvar.237489 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.237451 _fvar.237452, _fvar.237485, _fvar.237486, _fvar.237487, _fvar.237491, _fvar.237492 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.237488 (_fvar.237488 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.237488 (_fvar.237488 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.237488 (_fvar.237488 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.237488 (_fvar.237488 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.237451 _fvar.237452, _fvar.237485, _fvar.237486, _fvar.237487, _fvar.237491, _fvar.237492 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.237489 - _fvar.208949) _fvar.237489 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.237489 - _fvar.208949) _fvar.237489 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.237489 - _fvar.208949) _fvar.237489 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.237489 - _fvar.208949) _fvar.237489Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.237451 _fvar.237452, _fvar.237485, _fvar.237486, _fvar.237487, _fvar.237491, _fvar.237492 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.237488 (_fvar.237489 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.237488 (_fvar.237489 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.237488 (_fvar.237489 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.237488 (_fvar.237489 - _fvar.208949)Ileftlen:Ileft.length = ε⊢ Iright.length = εf:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Icc a✝ b✝)hf:ContinuousOn f ↑(Icc a✝ b✝)hI:(↑(Icc a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Icc _fvar.237503 _fvar.237504).ab:ℝ := (Chapter11.BoundedInterval.Icc _fvar.237503 _fvar.237504).bhsing:a < blower_le_upper:lower_integral f (Icc a✝ b✝) ≤ upper_integral f (Icc a✝ b✝)hM:∀ x ∈ ↑(Icc a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.237540 + _fvar.208949) (_fvar.237541 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.237503 _fvar.237504, _fvar.237537, _fvar.237538, _fvar.237539, _fvar.237543, _fvar.237544 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.237540 (_fvar.237540 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.237540 (_fvar.237540 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.237540 (_fvar.237540 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.237540 (_fvar.237540 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.237503 _fvar.237504, _fvar.237537, _fvar.237538, _fvar.237539, _fvar.237543, _fvar.237544 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.237541 - _fvar.208949) _fvar.237541 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.237541 - _fvar.208949) _fvar.237541 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.237541 - _fvar.208949) _fvar.237541 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.237541 - _fvar.208949) _fvar.237541Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.237503 _fvar.237504, _fvar.237537, _fvar.237538, _fvar.237539, _fvar.237543, _fvar.237544 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.237540 (_fvar.237541 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.237540 (_fvar.237541 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.237540 (_fvar.237541 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.237540 (_fvar.237541 - _fvar.208949)Ileftlen:Ileft.length = ε⊢ Iright.length = εf:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ioc a✝ b✝)hf:ContinuousOn f ↑(Ioc a✝ b✝)hI:(↑(Ioc a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ioc _fvar.237555 _fvar.237556).ab:ℝ := (Chapter11.BoundedInterval.Ioc _fvar.237555 _fvar.237556).bhsing:a < blower_le_upper:lower_integral f (Ioc a✝ b✝) ≤ upper_integral f (Ioc a✝ b✝)hM:∀ x ∈ ↑(Ioc a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.237592 + _fvar.208949) (_fvar.237593 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.237555 _fvar.237556, _fvar.237589, _fvar.237590, _fvar.237591, _fvar.237595, _fvar.237596 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.237592 (_fvar.237592 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.237592 (_fvar.237592 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.237592 (_fvar.237592 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.237592 (_fvar.237592 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.237555 _fvar.237556, _fvar.237589, _fvar.237590, _fvar.237591, _fvar.237595, _fvar.237596 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.237593 - _fvar.208949) _fvar.237593 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.237593 - _fvar.208949) _fvar.237593 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.237593 - _fvar.208949) _fvar.237593 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.237593 - _fvar.208949) _fvar.237593Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.237555 _fvar.237556, _fvar.237589, _fvar.237590, _fvar.237591, _fvar.237595, _fvar.237596 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.237592 (_fvar.237593 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.237592 (_fvar.237593 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.237592 (_fvar.237593 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.237592 (_fvar.237593 - _fvar.208949)Ileftlen:Ileft.length = ε⊢ Iright.length = εf:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ico a✝ b✝)hf:ContinuousOn f ↑(Ico a✝ b✝)hI:(↑(Ico a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ico _fvar.237607 _fvar.237608).ab:ℝ := (Chapter11.BoundedInterval.Ico _fvar.237607 _fvar.237608).bhsing:a < blower_le_upper:lower_integral f (Ico a✝ b✝) ≤ upper_integral f (Ico a✝ b✝)hM:∀ x ∈ ↑(Ico a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.237644 + _fvar.208949) (_fvar.237645 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.237607 _fvar.237608, _fvar.237641, _fvar.237642, _fvar.237643, _fvar.237647, _fvar.237648 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.237644 (_fvar.237644 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.237644 (_fvar.237644 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.237644 (_fvar.237644 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.237644 (_fvar.237644 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.237607 _fvar.237608, _fvar.237641, _fvar.237642, _fvar.237643, _fvar.237647, _fvar.237648 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.237645 - _fvar.208949) _fvar.237645 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.237645 - _fvar.208949) _fvar.237645 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.237645 - _fvar.208949) _fvar.237645 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.237645 - _fvar.208949) _fvar.237645Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.237607 _fvar.237608, _fvar.237641, _fvar.237642, _fvar.237643, _fvar.237647, _fvar.237648 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.237644 (_fvar.237645 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.237644 (_fvar.237645 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.237644 (_fvar.237645 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.237644 (_fvar.237645 - _fvar.208949)Ileftlen:Ileft.length = ε⊢ Iright.length = ε f:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ioo a✝ b✝)hf:ContinuousOn f ↑(Ioo a✝ b✝)hI:(↑(Ioo a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ioo _fvar.237451 _fvar.237452).ab:ℝ := (Chapter11.BoundedInterval.Ioo _fvar.237451 _fvar.237452).bhsing:a < blower_le_upper:lower_integral f (Ioo a✝ b✝) ≤ upper_integral f (Ioo a✝ b✝)hM:∀ x ∈ ↑(Ioo a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.237488 + _fvar.208949) (_fvar.237489 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.237451 _fvar.237452, _fvar.237485, _fvar.237486, _fvar.237487, _fvar.237491, _fvar.237492 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.237488 (_fvar.237488 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.237488 (_fvar.237488 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.237488 (_fvar.237488 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.237488 (_fvar.237488 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.237451 _fvar.237452, _fvar.237485, _fvar.237486, _fvar.237487, _fvar.237491, _fvar.237492 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.237489 - _fvar.208949) _fvar.237489 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.237489 - _fvar.208949) _fvar.237489 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.237489 - _fvar.208949) _fvar.237489 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.237489 - _fvar.208949) _fvar.237489Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.237451 _fvar.237452, _fvar.237485, _fvar.237486, _fvar.237487, _fvar.237491, _fvar.237492 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.237488 (_fvar.237489 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.237488 (_fvar.237489 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.237488 (_fvar.237489 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.237488 (_fvar.237489 - _fvar.208949)Ileftlen:Ileft.length = ε⊢ Iright.length = εf:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Icc a✝ b✝)hf:ContinuousOn f ↑(Icc a✝ b✝)hI:(↑(Icc a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Icc _fvar.237503 _fvar.237504).ab:ℝ := (Chapter11.BoundedInterval.Icc _fvar.237503 _fvar.237504).bhsing:a < blower_le_upper:lower_integral f (Icc a✝ b✝) ≤ upper_integral f (Icc a✝ b✝)hM:∀ x ∈ ↑(Icc a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.237540 + _fvar.208949) (_fvar.237541 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.237503 _fvar.237504, _fvar.237537, _fvar.237538, _fvar.237539, _fvar.237543, _fvar.237544 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.237540 (_fvar.237540 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.237540 (_fvar.237540 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.237540 (_fvar.237540 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.237540 (_fvar.237540 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.237503 _fvar.237504, _fvar.237537, _fvar.237538, _fvar.237539, _fvar.237543, _fvar.237544 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.237541 - _fvar.208949) _fvar.237541 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.237541 - _fvar.208949) _fvar.237541 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.237541 - _fvar.208949) _fvar.237541 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.237541 - _fvar.208949) _fvar.237541Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.237503 _fvar.237504, _fvar.237537, _fvar.237538, _fvar.237539, _fvar.237543, _fvar.237544 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.237540 (_fvar.237541 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.237540 (_fvar.237541 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.237540 (_fvar.237541 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.237540 (_fvar.237541 - _fvar.208949)Ileftlen:Ileft.length = ε⊢ Iright.length = εf:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ioc a✝ b✝)hf:ContinuousOn f ↑(Ioc a✝ b✝)hI:(↑(Ioc a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ioc _fvar.237555 _fvar.237556).ab:ℝ := (Chapter11.BoundedInterval.Ioc _fvar.237555 _fvar.237556).bhsing:a < blower_le_upper:lower_integral f (Ioc a✝ b✝) ≤ upper_integral f (Ioc a✝ b✝)hM:∀ x ∈ ↑(Ioc a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.237592 + _fvar.208949) (_fvar.237593 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.237555 _fvar.237556, _fvar.237589, _fvar.237590, _fvar.237591, _fvar.237595, _fvar.237596 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.237592 (_fvar.237592 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.237592 (_fvar.237592 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.237592 (_fvar.237592 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.237592 (_fvar.237592 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.237555 _fvar.237556, _fvar.237589, _fvar.237590, _fvar.237591, _fvar.237595, _fvar.237596 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.237593 - _fvar.208949) _fvar.237593 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.237593 - _fvar.208949) _fvar.237593 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.237593 - _fvar.208949) _fvar.237593 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.237593 - _fvar.208949) _fvar.237593Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.237555 _fvar.237556, _fvar.237589, _fvar.237590, _fvar.237591, _fvar.237595, _fvar.237596 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.237592 (_fvar.237593 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.237592 (_fvar.237593 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.237592 (_fvar.237593 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.237592 (_fvar.237593 - _fvar.208949)Ileftlen:Ileft.length = ε⊢ Iright.length = εf:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ico a✝ b✝)hf:ContinuousOn f ↑(Ico a✝ b✝)hI:(↑(Ico a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ico _fvar.237607 _fvar.237608).ab:ℝ := (Chapter11.BoundedInterval.Ico _fvar.237607 _fvar.237608).bhsing:a < blower_le_upper:lower_integral f (Ico a✝ b✝) ≤ upper_integral f (Ico a✝ b✝)hM:∀ x ∈ ↑(Ico a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.237644 + _fvar.208949) (_fvar.237645 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.237607 _fvar.237608, _fvar.237641, _fvar.237642, _fvar.237643, _fvar.237647, _fvar.237648 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.237644 (_fvar.237644 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.237644 (_fvar.237644 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.237644 (_fvar.237644 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.237644 (_fvar.237644 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.237607 _fvar.237608, _fvar.237641, _fvar.237642, _fvar.237643, _fvar.237647, _fvar.237648 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.237645 - _fvar.208949) _fvar.237645 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.237645 - _fvar.208949) _fvar.237645 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.237645 - _fvar.208949) _fvar.237645 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.237645 - _fvar.208949) _fvar.237645Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.237607 _fvar.237608, _fvar.237641, _fvar.237642, _fvar.237643, _fvar.237647, _fvar.237648 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.237644 (_fvar.237645 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.237644 (_fvar.237645 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.237644 (_fvar.237645 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.237644 (_fvar.237645 - _fvar.208949)Ileftlen:Ileft.length = ε⊢ Iright.length = ε All goals completed! 🐙 have hjoin1 : Ileft'.joins Ileft I' := I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑I⊢ IntegrableOn f I f:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ioo a✝ b✝)hf:ContinuousOn f ↑(Ioo a✝ b✝)hI:(↑(Ioo a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520).ab:ℝ := (Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520).bhsing:a < blower_le_upper:lower_integral f (Ioo a✝ b✝) ≤ upper_integral f (Ioo a✝ b✝)hM:∀ x ∈ ↑(Ioo a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.241557 + _fvar.208949) (_fvar.241558 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520, _fvar.241554, _fvar.241555, _fvar.241556, _fvar.241560, _fvar.241561 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241557 (_fvar.241557 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241557 (_fvar.241557 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241557 (_fvar.241557 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241557 (_fvar.241557 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520, _fvar.241554, _fvar.241555, _fvar.241556, _fvar.241560, _fvar.241561 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241558 - _fvar.208949) _fvar.241558 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241558 - _fvar.208949) _fvar.241558 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241558 - _fvar.208949) _fvar.241558 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241558 - _fvar.208949) _fvar.241558Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520, _fvar.241554, _fvar.241555, _fvar.241556, _fvar.241560, _fvar.241561 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241557 (_fvar.241558 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241557 (_fvar.241558 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241557 (_fvar.241558 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241557 (_fvar.241558 - _fvar.208949)Ileftlen:Ileft.length = εIrightlen:Iright.length = ε⊢ Ileft'.joins Ileft I'f:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Icc a✝ b✝)hf:ContinuousOn f ↑(Icc a✝ b✝)hI:(↑(Icc a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574).ab:ℝ := (Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574).bhsing:a < blower_le_upper:lower_integral f (Icc a✝ b✝) ≤ upper_integral f (Icc a✝ b✝)hM:∀ x ∈ ↑(Icc a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.241611 + _fvar.208949) (_fvar.241612 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574, _fvar.241608, _fvar.241609, _fvar.241610, _fvar.241614, _fvar.241615 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241611 (_fvar.241611 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241611 (_fvar.241611 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241611 (_fvar.241611 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241611 (_fvar.241611 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574, _fvar.241608, _fvar.241609, _fvar.241610, _fvar.241614, _fvar.241615 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241612 - _fvar.208949) _fvar.241612 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241612 - _fvar.208949) _fvar.241612 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241612 - _fvar.208949) _fvar.241612 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241612 - _fvar.208949) _fvar.241612Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574, _fvar.241608, _fvar.241609, _fvar.241610, _fvar.241614, _fvar.241615 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241611 (_fvar.241612 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241611 (_fvar.241612 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241611 (_fvar.241612 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241611 (_fvar.241612 - _fvar.208949)Ileftlen:Ileft.length = εIrightlen:Iright.length = ε⊢ Ileft'.joins Ileft I'f:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ioc a✝ b✝)hf:ContinuousOn f ↑(Ioc a✝ b✝)hI:(↑(Ioc a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628).ab:ℝ := (Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628).bhsing:a < blower_le_upper:lower_integral f (Ioc a✝ b✝) ≤ upper_integral f (Ioc a✝ b✝)hM:∀ x ∈ ↑(Ioc a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.241665 + _fvar.208949) (_fvar.241666 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628, _fvar.241662, _fvar.241663, _fvar.241664, _fvar.241668, _fvar.241669 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241665 (_fvar.241665 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241665 (_fvar.241665 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241665 (_fvar.241665 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241665 (_fvar.241665 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628, _fvar.241662, _fvar.241663, _fvar.241664, _fvar.241668, _fvar.241669 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241666 - _fvar.208949) _fvar.241666 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241666 - _fvar.208949) _fvar.241666 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241666 - _fvar.208949) _fvar.241666 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241666 - _fvar.208949) _fvar.241666Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628, _fvar.241662, _fvar.241663, _fvar.241664, _fvar.241668, _fvar.241669 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241665 (_fvar.241666 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241665 (_fvar.241666 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241665 (_fvar.241666 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241665 (_fvar.241666 - _fvar.208949)Ileftlen:Ileft.length = εIrightlen:Iright.length = ε⊢ Ileft'.joins Ileft I'f:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ico a✝ b✝)hf:ContinuousOn f ↑(Ico a✝ b✝)hI:(↑(Ico a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682).ab:ℝ := (Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682).bhsing:a < blower_le_upper:lower_integral f (Ico a✝ b✝) ≤ upper_integral f (Ico a✝ b✝)hM:∀ x ∈ ↑(Ico a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.241719 + _fvar.208949) (_fvar.241720 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682, _fvar.241716, _fvar.241717, _fvar.241718, _fvar.241722, _fvar.241723 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241719 (_fvar.241719 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241719 (_fvar.241719 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241719 (_fvar.241719 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241719 (_fvar.241719 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682, _fvar.241716, _fvar.241717, _fvar.241718, _fvar.241722, _fvar.241723 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241720 - _fvar.208949) _fvar.241720 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241720 - _fvar.208949) _fvar.241720 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241720 - _fvar.208949) _fvar.241720 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241720 - _fvar.208949) _fvar.241720Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682, _fvar.241716, _fvar.241717, _fvar.241718, _fvar.241722, _fvar.241723 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241719 (_fvar.241720 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241719 (_fvar.241720 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241719 (_fvar.241720 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241719 (_fvar.241720 - _fvar.208949)Ileftlen:Ileft.length = εIrightlen:Iright.length = ε⊢ Ileft'.joins Ileft I' case Icc _ _ f:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Icc a✝ b✝)hf:ContinuousOn f ↑(Icc a✝ b✝)hI:(↑(Icc a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574).ab:ℝ := (Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574).bhsing:a < blower_le_upper:lower_integral f (Icc a✝ b✝) ≤ upper_integral f (Icc a✝ b✝)hM:∀ x ∈ ↑(Icc a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.241611 + _fvar.208949) (_fvar.241612 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574, _fvar.241608, _fvar.241609, _fvar.241610, _fvar.241614, _fvar.241615 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241611 (_fvar.241611 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241611 (_fvar.241611 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241611 (_fvar.241611 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241611 (_fvar.241611 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574, _fvar.241608, _fvar.241609, _fvar.241610, _fvar.241614, _fvar.241615 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241612 - _fvar.208949) _fvar.241612 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241612 - _fvar.208949) _fvar.241612 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241612 - _fvar.208949) _fvar.241612 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241612 - _fvar.208949) _fvar.241612Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574, _fvar.241608, _fvar.241609, _fvar.241610, _fvar.241614, _fvar.241615 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241611 (_fvar.241612 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241611 (_fvar.241612 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241611 (_fvar.241612 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241611 (_fvar.241612 - _fvar.208949)Ileftlen:Ileft.length = εIrightlen:Iright.length = ε⊢ Ileft'.joins Ileft I' f:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Icc a✝ b✝)hf:ContinuousOn f ↑(Icc a✝ b✝)hI:(↑(Icc a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574).ab:ℝ := (Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574).bhsing:a < blower_le_upper:lower_integral f (Icc a✝ b✝) ≤ upper_integral f (Icc a✝ b✝)hM:∀ x ∈ ↑(Icc a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.241611 + _fvar.208949) (_fvar.241612 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574, _fvar.241608, _fvar.241609, _fvar.241610, _fvar.241614, _fvar.241615 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241611 (_fvar.241611 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241611 (_fvar.241611 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241611 (_fvar.241611 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241611 (_fvar.241611 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574, _fvar.241608, _fvar.241609, _fvar.241610, _fvar.241614, _fvar.241615 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241612 - _fvar.208949) _fvar.241612 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241612 - _fvar.208949) _fvar.241612 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241612 - _fvar.208949) _fvar.241612 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241612 - _fvar.208949) _fvar.241612Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574, _fvar.241608, _fvar.241609, _fvar.241610, _fvar.241614, _fvar.241615 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241611 (_fvar.241612 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241611 (_fvar.241612 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241611 (_fvar.241612 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241611 (_fvar.241612 - _fvar.208949)Ileftlen:Ileft.length = εIrightlen:Iright.length = ε⊢ a ≤ a + εf:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Icc a✝ b✝)hf:ContinuousOn f ↑(Icc a✝ b✝)hI:(↑(Icc a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574).ab:ℝ := (Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574).bhsing:a < blower_le_upper:lower_integral f (Icc a✝ b✝) ≤ upper_integral f (Icc a✝ b✝)hM:∀ x ∈ ↑(Icc a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.241611 + _fvar.208949) (_fvar.241612 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574, _fvar.241608, _fvar.241609, _fvar.241610, _fvar.241614, _fvar.241615 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241611 (_fvar.241611 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241611 (_fvar.241611 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241611 (_fvar.241611 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241611 (_fvar.241611 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574, _fvar.241608, _fvar.241609, _fvar.241610, _fvar.241614, _fvar.241615 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241612 - _fvar.208949) _fvar.241612 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241612 - _fvar.208949) _fvar.241612 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241612 - _fvar.208949) _fvar.241612 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241612 - _fvar.208949) _fvar.241612Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574, _fvar.241608, _fvar.241609, _fvar.241610, _fvar.241614, _fvar.241615 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241611 (_fvar.241612 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241611 (_fvar.241612 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241611 (_fvar.241612 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241611 (_fvar.241612 - _fvar.208949)Ileftlen:Ileft.length = εIrightlen:Iright.length = ε⊢ a + ε ≤ b - ε f:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Icc a✝ b✝)hf:ContinuousOn f ↑(Icc a✝ b✝)hI:(↑(Icc a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574).ab:ℝ := (Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574).bhsing:a < blower_le_upper:lower_integral f (Icc a✝ b✝) ≤ upper_integral f (Icc a✝ b✝)hM:∀ x ∈ ↑(Icc a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.241611 + _fvar.208949) (_fvar.241612 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574, _fvar.241608, _fvar.241609, _fvar.241610, _fvar.241614, _fvar.241615 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241611 (_fvar.241611 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241611 (_fvar.241611 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241611 (_fvar.241611 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241611 (_fvar.241611 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574, _fvar.241608, _fvar.241609, _fvar.241610, _fvar.241614, _fvar.241615 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241612 - _fvar.208949) _fvar.241612 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241612 - _fvar.208949) _fvar.241612 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241612 - _fvar.208949) _fvar.241612 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241612 - _fvar.208949) _fvar.241612Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574, _fvar.241608, _fvar.241609, _fvar.241610, _fvar.241614, _fvar.241615 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241611 (_fvar.241612 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241611 (_fvar.241612 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241611 (_fvar.241612 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241611 (_fvar.241612 - _fvar.208949)Ileftlen:Ileft.length = εIrightlen:Iright.length = ε⊢ a ≤ a + εf:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Icc a✝ b✝)hf:ContinuousOn f ↑(Icc a✝ b✝)hI:(↑(Icc a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574).ab:ℝ := (Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574).bhsing:a < blower_le_upper:lower_integral f (Icc a✝ b✝) ≤ upper_integral f (Icc a✝ b✝)hM:∀ x ∈ ↑(Icc a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.241611 + _fvar.208949) (_fvar.241612 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574, _fvar.241608, _fvar.241609, _fvar.241610, _fvar.241614, _fvar.241615 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241611 (_fvar.241611 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241611 (_fvar.241611 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241611 (_fvar.241611 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241611 (_fvar.241611 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574, _fvar.241608, _fvar.241609, _fvar.241610, _fvar.241614, _fvar.241615 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241612 - _fvar.208949) _fvar.241612 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241612 - _fvar.208949) _fvar.241612 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241612 - _fvar.208949) _fvar.241612 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241612 - _fvar.208949) _fvar.241612Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Icc _fvar.241573 _fvar.241574, _fvar.241608, _fvar.241609, _fvar.241610, _fvar.241614, _fvar.241615 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241611 (_fvar.241612 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241611 (_fvar.241612 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241611 (_fvar.241612 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241611 (_fvar.241612 - _fvar.208949)Ileftlen:Ileft.length = εIrightlen:Iright.length = ε⊢ a + ε ≤ b - ε All goals completed! 🐙 case Ico _ _ f:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ico a✝ b✝)hf:ContinuousOn f ↑(Ico a✝ b✝)hI:(↑(Ico a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682).ab:ℝ := (Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682).bhsing:a < blower_le_upper:lower_integral f (Ico a✝ b✝) ≤ upper_integral f (Ico a✝ b✝)hM:∀ x ∈ ↑(Ico a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.241719 + _fvar.208949) (_fvar.241720 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682, _fvar.241716, _fvar.241717, _fvar.241718, _fvar.241722, _fvar.241723 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241719 (_fvar.241719 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241719 (_fvar.241719 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241719 (_fvar.241719 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241719 (_fvar.241719 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682, _fvar.241716, _fvar.241717, _fvar.241718, _fvar.241722, _fvar.241723 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241720 - _fvar.208949) _fvar.241720 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241720 - _fvar.208949) _fvar.241720 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241720 - _fvar.208949) _fvar.241720 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241720 - _fvar.208949) _fvar.241720Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682, _fvar.241716, _fvar.241717, _fvar.241718, _fvar.241722, _fvar.241723 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241719 (_fvar.241720 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241719 (_fvar.241720 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241719 (_fvar.241720 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241719 (_fvar.241720 - _fvar.208949)Ileftlen:Ileft.length = εIrightlen:Iright.length = ε⊢ Ileft'.joins Ileft I' f:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ico a✝ b✝)hf:ContinuousOn f ↑(Ico a✝ b✝)hI:(↑(Ico a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682).ab:ℝ := (Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682).bhsing:a < blower_le_upper:lower_integral f (Ico a✝ b✝) ≤ upper_integral f (Ico a✝ b✝)hM:∀ x ∈ ↑(Ico a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.241719 + _fvar.208949) (_fvar.241720 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682, _fvar.241716, _fvar.241717, _fvar.241718, _fvar.241722, _fvar.241723 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241719 (_fvar.241719 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241719 (_fvar.241719 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241719 (_fvar.241719 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241719 (_fvar.241719 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682, _fvar.241716, _fvar.241717, _fvar.241718, _fvar.241722, _fvar.241723 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241720 - _fvar.208949) _fvar.241720 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241720 - _fvar.208949) _fvar.241720 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241720 - _fvar.208949) _fvar.241720 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241720 - _fvar.208949) _fvar.241720Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682, _fvar.241716, _fvar.241717, _fvar.241718, _fvar.241722, _fvar.241723 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241719 (_fvar.241720 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241719 (_fvar.241720 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241719 (_fvar.241720 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241719 (_fvar.241720 - _fvar.208949)Ileftlen:Ileft.length = εIrightlen:Iright.length = ε⊢ a ≤ a + εf:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ico a✝ b✝)hf:ContinuousOn f ↑(Ico a✝ b✝)hI:(↑(Ico a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682).ab:ℝ := (Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682).bhsing:a < blower_le_upper:lower_integral f (Ico a✝ b✝) ≤ upper_integral f (Ico a✝ b✝)hM:∀ x ∈ ↑(Ico a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.241719 + _fvar.208949) (_fvar.241720 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682, _fvar.241716, _fvar.241717, _fvar.241718, _fvar.241722, _fvar.241723 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241719 (_fvar.241719 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241719 (_fvar.241719 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241719 (_fvar.241719 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241719 (_fvar.241719 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682, _fvar.241716, _fvar.241717, _fvar.241718, _fvar.241722, _fvar.241723 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241720 - _fvar.208949) _fvar.241720 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241720 - _fvar.208949) _fvar.241720 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241720 - _fvar.208949) _fvar.241720 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241720 - _fvar.208949) _fvar.241720Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682, _fvar.241716, _fvar.241717, _fvar.241718, _fvar.241722, _fvar.241723 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241719 (_fvar.241720 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241719 (_fvar.241720 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241719 (_fvar.241720 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241719 (_fvar.241720 - _fvar.208949)Ileftlen:Ileft.length = εIrightlen:Iright.length = ε⊢ a + ε ≤ b - ε f:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ico a✝ b✝)hf:ContinuousOn f ↑(Ico a✝ b✝)hI:(↑(Ico a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682).ab:ℝ := (Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682).bhsing:a < blower_le_upper:lower_integral f (Ico a✝ b✝) ≤ upper_integral f (Ico a✝ b✝)hM:∀ x ∈ ↑(Ico a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.241719 + _fvar.208949) (_fvar.241720 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682, _fvar.241716, _fvar.241717, _fvar.241718, _fvar.241722, _fvar.241723 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241719 (_fvar.241719 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241719 (_fvar.241719 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241719 (_fvar.241719 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241719 (_fvar.241719 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682, _fvar.241716, _fvar.241717, _fvar.241718, _fvar.241722, _fvar.241723 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241720 - _fvar.208949) _fvar.241720 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241720 - _fvar.208949) _fvar.241720 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241720 - _fvar.208949) _fvar.241720 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241720 - _fvar.208949) _fvar.241720Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682, _fvar.241716, _fvar.241717, _fvar.241718, _fvar.241722, _fvar.241723 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241719 (_fvar.241720 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241719 (_fvar.241720 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241719 (_fvar.241720 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241719 (_fvar.241720 - _fvar.208949)Ileftlen:Ileft.length = εIrightlen:Iright.length = ε⊢ a ≤ a + εf:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ico a✝ b✝)hf:ContinuousOn f ↑(Ico a✝ b✝)hI:(↑(Ico a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682).ab:ℝ := (Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682).bhsing:a < blower_le_upper:lower_integral f (Ico a✝ b✝) ≤ upper_integral f (Ico a✝ b✝)hM:∀ x ∈ ↑(Ico a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.241719 + _fvar.208949) (_fvar.241720 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682, _fvar.241716, _fvar.241717, _fvar.241718, _fvar.241722, _fvar.241723 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241719 (_fvar.241719 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241719 (_fvar.241719 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241719 (_fvar.241719 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241719 (_fvar.241719 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682, _fvar.241716, _fvar.241717, _fvar.241718, _fvar.241722, _fvar.241723 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241720 - _fvar.208949) _fvar.241720 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241720 - _fvar.208949) _fvar.241720 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241720 - _fvar.208949) _fvar.241720 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241720 - _fvar.208949) _fvar.241720Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ico _fvar.241681 _fvar.241682, _fvar.241716, _fvar.241717, _fvar.241718, _fvar.241722, _fvar.241723 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241719 (_fvar.241720 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241719 (_fvar.241720 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241719 (_fvar.241720 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241719 (_fvar.241720 - _fvar.208949)Ileftlen:Ileft.length = εIrightlen:Iright.length = ε⊢ a + ε ≤ b - ε All goals completed! 🐙 case Ioc _ _ f:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ioc a✝ b✝)hf:ContinuousOn f ↑(Ioc a✝ b✝)hI:(↑(Ioc a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628).ab:ℝ := (Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628).bhsing:a < blower_le_upper:lower_integral f (Ioc a✝ b✝) ≤ upper_integral f (Ioc a✝ b✝)hM:∀ x ∈ ↑(Ioc a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.241665 + _fvar.208949) (_fvar.241666 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628, _fvar.241662, _fvar.241663, _fvar.241664, _fvar.241668, _fvar.241669 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241665 (_fvar.241665 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241665 (_fvar.241665 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241665 (_fvar.241665 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241665 (_fvar.241665 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628, _fvar.241662, _fvar.241663, _fvar.241664, _fvar.241668, _fvar.241669 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241666 - _fvar.208949) _fvar.241666 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241666 - _fvar.208949) _fvar.241666 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241666 - _fvar.208949) _fvar.241666 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241666 - _fvar.208949) _fvar.241666Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628, _fvar.241662, _fvar.241663, _fvar.241664, _fvar.241668, _fvar.241669 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241665 (_fvar.241666 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241665 (_fvar.241666 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241665 (_fvar.241666 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241665 (_fvar.241666 - _fvar.208949)Ileftlen:Ileft.length = εIrightlen:Iright.length = ε⊢ Ileft'.joins Ileft I' f:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ioc a✝ b✝)hf:ContinuousOn f ↑(Ioc a✝ b✝)hI:(↑(Ioc a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628).ab:ℝ := (Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628).bhsing:a < blower_le_upper:lower_integral f (Ioc a✝ b✝) ≤ upper_integral f (Ioc a✝ b✝)hM:∀ x ∈ ↑(Ioc a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.241665 + _fvar.208949) (_fvar.241666 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628, _fvar.241662, _fvar.241663, _fvar.241664, _fvar.241668, _fvar.241669 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241665 (_fvar.241665 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241665 (_fvar.241665 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241665 (_fvar.241665 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241665 (_fvar.241665 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628, _fvar.241662, _fvar.241663, _fvar.241664, _fvar.241668, _fvar.241669 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241666 - _fvar.208949) _fvar.241666 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241666 - _fvar.208949) _fvar.241666 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241666 - _fvar.208949) _fvar.241666 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241666 - _fvar.208949) _fvar.241666Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628, _fvar.241662, _fvar.241663, _fvar.241664, _fvar.241668, _fvar.241669 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241665 (_fvar.241666 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241665 (_fvar.241666 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241665 (_fvar.241666 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241665 (_fvar.241666 - _fvar.208949)Ileftlen:Ileft.length = εIrightlen:Iright.length = ε⊢ a < a + εf:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ioc a✝ b✝)hf:ContinuousOn f ↑(Ioc a✝ b✝)hI:(↑(Ioc a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628).ab:ℝ := (Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628).bhsing:a < blower_le_upper:lower_integral f (Ioc a✝ b✝) ≤ upper_integral f (Ioc a✝ b✝)hM:∀ x ∈ ↑(Ioc a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.241665 + _fvar.208949) (_fvar.241666 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628, _fvar.241662, _fvar.241663, _fvar.241664, _fvar.241668, _fvar.241669 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241665 (_fvar.241665 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241665 (_fvar.241665 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241665 (_fvar.241665 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241665 (_fvar.241665 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628, _fvar.241662, _fvar.241663, _fvar.241664, _fvar.241668, _fvar.241669 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241666 - _fvar.208949) _fvar.241666 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241666 - _fvar.208949) _fvar.241666 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241666 - _fvar.208949) _fvar.241666 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241666 - _fvar.208949) _fvar.241666Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628, _fvar.241662, _fvar.241663, _fvar.241664, _fvar.241668, _fvar.241669 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241665 (_fvar.241666 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241665 (_fvar.241666 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241665 (_fvar.241666 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241665 (_fvar.241666 - _fvar.208949)Ileftlen:Ileft.length = εIrightlen:Iright.length = ε⊢ a + ε ≤ b - ε f:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ioc a✝ b✝)hf:ContinuousOn f ↑(Ioc a✝ b✝)hI:(↑(Ioc a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628).ab:ℝ := (Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628).bhsing:a < blower_le_upper:lower_integral f (Ioc a✝ b✝) ≤ upper_integral f (Ioc a✝ b✝)hM:∀ x ∈ ↑(Ioc a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.241665 + _fvar.208949) (_fvar.241666 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628, _fvar.241662, _fvar.241663, _fvar.241664, _fvar.241668, _fvar.241669 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241665 (_fvar.241665 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241665 (_fvar.241665 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241665 (_fvar.241665 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241665 (_fvar.241665 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628, _fvar.241662, _fvar.241663, _fvar.241664, _fvar.241668, _fvar.241669 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241666 - _fvar.208949) _fvar.241666 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241666 - _fvar.208949) _fvar.241666 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241666 - _fvar.208949) _fvar.241666 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241666 - _fvar.208949) _fvar.241666Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628, _fvar.241662, _fvar.241663, _fvar.241664, _fvar.241668, _fvar.241669 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241665 (_fvar.241666 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241665 (_fvar.241666 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241665 (_fvar.241666 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241665 (_fvar.241666 - _fvar.208949)Ileftlen:Ileft.length = εIrightlen:Iright.length = ε⊢ a < a + εf:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ioc a✝ b✝)hf:ContinuousOn f ↑(Ioc a✝ b✝)hI:(↑(Ioc a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628).ab:ℝ := (Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628).bhsing:a < blower_le_upper:lower_integral f (Ioc a✝ b✝) ≤ upper_integral f (Ioc a✝ b✝)hM:∀ x ∈ ↑(Ioc a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.241665 + _fvar.208949) (_fvar.241666 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628, _fvar.241662, _fvar.241663, _fvar.241664, _fvar.241668, _fvar.241669 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241665 (_fvar.241665 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241665 (_fvar.241665 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241665 (_fvar.241665 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241665 (_fvar.241665 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628, _fvar.241662, _fvar.241663, _fvar.241664, _fvar.241668, _fvar.241669 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241666 - _fvar.208949) _fvar.241666 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241666 - _fvar.208949) _fvar.241666 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241666 - _fvar.208949) _fvar.241666 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241666 - _fvar.208949) _fvar.241666Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioc _fvar.241627 _fvar.241628, _fvar.241662, _fvar.241663, _fvar.241664, _fvar.241668, _fvar.241669 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241665 (_fvar.241666 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241665 (_fvar.241666 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241665 (_fvar.241666 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241665 (_fvar.241666 - _fvar.208949)Ileftlen:Ileft.length = εIrightlen:Iright.length = ε⊢ a + ε ≤ b - ε All goals completed! 🐙 case Ioo _ _ f:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ioo a✝ b✝)hf:ContinuousOn f ↑(Ioo a✝ b✝)hI:(↑(Ioo a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520).ab:ℝ := (Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520).bhsing:a < blower_le_upper:lower_integral f (Ioo a✝ b✝) ≤ upper_integral f (Ioo a✝ b✝)hM:∀ x ∈ ↑(Ioo a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.241557 + _fvar.208949) (_fvar.241558 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520, _fvar.241554, _fvar.241555, _fvar.241556, _fvar.241560, _fvar.241561 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241557 (_fvar.241557 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241557 (_fvar.241557 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241557 (_fvar.241557 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241557 (_fvar.241557 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520, _fvar.241554, _fvar.241555, _fvar.241556, _fvar.241560, _fvar.241561 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241558 - _fvar.208949) _fvar.241558 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241558 - _fvar.208949) _fvar.241558 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241558 - _fvar.208949) _fvar.241558 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241558 - _fvar.208949) _fvar.241558Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520, _fvar.241554, _fvar.241555, _fvar.241556, _fvar.241560, _fvar.241561 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241557 (_fvar.241558 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241557 (_fvar.241558 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241557 (_fvar.241558 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241557 (_fvar.241558 - _fvar.208949)Ileftlen:Ileft.length = εIrightlen:Iright.length = ε⊢ Ileft'.joins Ileft I' f:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ioo a✝ b✝)hf:ContinuousOn f ↑(Ioo a✝ b✝)hI:(↑(Ioo a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520).ab:ℝ := (Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520).bhsing:a < blower_le_upper:lower_integral f (Ioo a✝ b✝) ≤ upper_integral f (Ioo a✝ b✝)hM:∀ x ∈ ↑(Ioo a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.241557 + _fvar.208949) (_fvar.241558 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520, _fvar.241554, _fvar.241555, _fvar.241556, _fvar.241560, _fvar.241561 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241557 (_fvar.241557 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241557 (_fvar.241557 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241557 (_fvar.241557 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241557 (_fvar.241557 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520, _fvar.241554, _fvar.241555, _fvar.241556, _fvar.241560, _fvar.241561 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241558 - _fvar.208949) _fvar.241558 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241558 - _fvar.208949) _fvar.241558 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241558 - _fvar.208949) _fvar.241558 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241558 - _fvar.208949) _fvar.241558Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520, _fvar.241554, _fvar.241555, _fvar.241556, _fvar.241560, _fvar.241561 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241557 (_fvar.241558 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241557 (_fvar.241558 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241557 (_fvar.241558 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241557 (_fvar.241558 - _fvar.208949)Ileftlen:Ileft.length = εIrightlen:Iright.length = ε⊢ a < a + εf:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ioo a✝ b✝)hf:ContinuousOn f ↑(Ioo a✝ b✝)hI:(↑(Ioo a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520).ab:ℝ := (Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520).bhsing:a < blower_le_upper:lower_integral f (Ioo a✝ b✝) ≤ upper_integral f (Ioo a✝ b✝)hM:∀ x ∈ ↑(Ioo a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.241557 + _fvar.208949) (_fvar.241558 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520, _fvar.241554, _fvar.241555, _fvar.241556, _fvar.241560, _fvar.241561 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241557 (_fvar.241557 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241557 (_fvar.241557 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241557 (_fvar.241557 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241557 (_fvar.241557 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520, _fvar.241554, _fvar.241555, _fvar.241556, _fvar.241560, _fvar.241561 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241558 - _fvar.208949) _fvar.241558 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241558 - _fvar.208949) _fvar.241558 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241558 - _fvar.208949) _fvar.241558 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241558 - _fvar.208949) _fvar.241558Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520, _fvar.241554, _fvar.241555, _fvar.241556, _fvar.241560, _fvar.241561 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241557 (_fvar.241558 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241557 (_fvar.241558 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241557 (_fvar.241558 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241557 (_fvar.241558 - _fvar.208949)Ileftlen:Ileft.length = εIrightlen:Iright.length = ε⊢ a + ε ≤ b - ε f:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ioo a✝ b✝)hf:ContinuousOn f ↑(Ioo a✝ b✝)hI:(↑(Ioo a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520).ab:ℝ := (Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520).bhsing:a < blower_le_upper:lower_integral f (Ioo a✝ b✝) ≤ upper_integral f (Ioo a✝ b✝)hM:∀ x ∈ ↑(Ioo a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.241557 + _fvar.208949) (_fvar.241558 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520, _fvar.241554, _fvar.241555, _fvar.241556, _fvar.241560, _fvar.241561 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241557 (_fvar.241557 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241557 (_fvar.241557 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241557 (_fvar.241557 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241557 (_fvar.241557 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520, _fvar.241554, _fvar.241555, _fvar.241556, _fvar.241560, _fvar.241561 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241558 - _fvar.208949) _fvar.241558 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241558 - _fvar.208949) _fvar.241558 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241558 - _fvar.208949) _fvar.241558 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241558 - _fvar.208949) _fvar.241558Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520, _fvar.241554, _fvar.241555, _fvar.241556, _fvar.241560, _fvar.241561 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241557 (_fvar.241558 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241557 (_fvar.241558 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241557 (_fvar.241558 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241557 (_fvar.241558 - _fvar.208949)Ileftlen:Ileft.length = εIrightlen:Iright.length = ε⊢ a < a + εf:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ioo a✝ b✝)hf:ContinuousOn f ↑(Ioo a✝ b✝)hI:(↑(Ioo a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520).ab:ℝ := (Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520).bhsing:a < blower_le_upper:lower_integral f (Ioo a✝ b✝) ≤ upper_integral f (Ioo a✝ b✝)hM:∀ x ∈ ↑(Ioo a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.241557 + _fvar.208949) (_fvar.241558 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520, _fvar.241554, _fvar.241555, _fvar.241556, _fvar.241560, _fvar.241561 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241557 (_fvar.241557 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.241557 (_fvar.241557 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241557 (_fvar.241557 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.241557 (_fvar.241557 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520, _fvar.241554, _fvar.241555, _fvar.241556, _fvar.241560, _fvar.241561 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241558 - _fvar.208949) _fvar.241558 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241558 - _fvar.208949) _fvar.241558 | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.241558 - _fvar.208949) _fvar.241558 | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.241558 - _fvar.208949) _fvar.241558Ileft':Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.241519 _fvar.241520, _fvar.241554, _fvar.241555, _fvar.241556, _fvar.241560, _fvar.241561 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241557 (_fvar.241558 - _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Icc _fvar.241557 (_fvar.241558 - _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241557 (_fvar.241558 - _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc _fvar.241557 (_fvar.241558 - _fvar.208949)Ileftlen:Ileft.length = εIrightlen:Iright.length = ε⊢ a + ε ≤ b - ε All goals completed! 🐙 have hjoin2: I.joins Ileft' Iright := I:BoundedIntervalf:ℝ → ℝhbound:BddOn f ↑Ihf:ContinuousOn f ↑I⊢ IntegrableOn f I f:ℝ → ℝM:ℝhMpos:0 ≤ Mε:ℝhε:ε > 0a✝:ℝb✝:ℝhbound:BddOn f ↑(Ioo a✝ b✝)hf:ContinuousOn f ↑(Ioo a✝ b✝)hI:(↑(Ioo a✝ b✝)).Nonemptya:ℝ := (Chapter11.BoundedInterval.Ioo _fvar.251736 _fvar.251737).ab:ℝ := (Chapter11.BoundedInterval.Ioo _fvar.251736 _fvar.251737).bhsing:a < blower_le_upper:lower_integral f (Ioo a✝ b✝) ≤ upper_integral f (Ioo a✝ b✝)hM:∀ x ∈ ↑(Ioo a✝ b✝), |f x| ≤ Mhε':ε < (b - a) / 2I':Chapter11.BoundedInterval := Chapter11.BoundedInterval.Icc (_fvar.251775 + _fvar.208949) (_fvar.251776 - _fvar.208949)Ileft:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.251736 _fvar.251737, _fvar.251772, _fvar.251773, _fvar.251774, _fvar.251778, _fvar.251779 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.251775 (_fvar.251775 + _fvar.208949) | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ico _fvar.251775 (_fvar.251775 + _fvar.208949) | Chapter11.BoundedInterval.Ioc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.251775 (_fvar.251775 + _fvar.208949) | Chapter11.BoundedInterval.Ioo a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo _fvar.251775 (_fvar.251775 + _fvar.208949)Iright:Chapter11.BoundedInterval := match Chapter11.BoundedInterval.Ioo _fvar.251736 _fvar.251737, _fvar.251772, _fvar.251773, _fvar.251774, _fvar.251778, _fvar.251779 with | Chapter11.BoundedInterval.Icc a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioc (_fvar.251776 - _fvar.208949) _fvar.251776 | Chapter11.BoundedInterval.Ico a b, hbound, hf, hI, lower_le_upper, hM => Chapter11.BoundedInterval.Ioo (_fvar.251776 - _fvar.208949) _fvar.251776 | Chapter11.BoundedInterval.Ioc a b,