ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>x : M<VAR>s : Set M<VAR>Is : SmoothManifoldWithCorners I M<VAR>I's : SmoothManifoldWithCorners I' M'<VAR>h : MDifferentiableAt I I' f x<VAR>hxs : UniqueMDiffWithinAt I s x<GOAL>mfderivWithin I I' f s x = mfderiv I I' f x
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>x : M<VAR>s t : Set M<VAR>Is : SmoothManifoldWithCorners I M<VAR>I's : SmoothManifoldWithCorners I' M'<VAR>st : s ⊆ t<VAR>hs : UniqueMDiffWithinAt I s x<VAR>h : MDifferentiableWithinAt I I' f t x<GOAL>mfderivWithin I I' f s x = mfderivWithin I I' f t x
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>x : M<VAR>s t : Set M<VAR>hst : s ⊆ t<VAR>h : MDifferentiableWithinAt I I' f t x<GOAL>MDifferentiableWithinAt I I' f s x
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>x : M<GOAL>MDifferentiableWithinAt I I' f Set.univ x ↔ MDifferentiableAt I I' f x
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>x : M<VAR>s t : Set M<VAR>ht : t ∈ 𝓝 x<GOAL>MDifferentiableWithinAt I I' f (s ∩ t) x ↔ MDifferentiableWithinAt I I' f s x
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>x : M<VAR>s t : Set M<VAR>ht : t ∈ 𝓝[s] x<GOAL>MDifferentiableWithinAt I I' f (s ∩ t) x ↔ MDifferentiableWithinAt I I' f s x
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>x : M<VAR>s : Set M<VAR>h : MDifferentiableAt I I' f x<GOAL>MDifferentiableWithinAt I I' f s x
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>x : M<VAR>s : Set M<VAR>h : MDifferentiableWithinAt I I' f s x<VAR>hs : s ∈ 𝓝 x<GOAL>MDifferentiableAt I I' f x
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>x : M<VAR>s : Set M<VAR>h : MDifferentiableOn I I' f s<VAR>hx : s ∈ 𝓝 x<GOAL>MDifferentiableAt I I' f x
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>s t : Set M<VAR>h : MDifferentiableOn I I' f t<VAR>st : s ⊆ t<GOAL>MDifferentiableOn I I' f s
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<GOAL>MDifferentiableOn I I' f Set.univ ↔ MDifferentiable I I' f
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>s : Set M<VAR>h : MDifferentiable I I' f<GOAL>MDifferentiableOn I I' f s
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>s : Set M<VAR>h : ∀ x ∈ s, ∃ u, IsOpen u ∧ x ∈ u ∧ MDifferentiableOn I I' f (s ∩ u)<GOAL>MDifferentiableOn I I' f s
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>Is : SmoothManifoldWithCorners I M<VAR>I's : SmoothManifoldWithCorners I' M'<GOAL>mfderivWithin I I' f Set.univ = mfderiv I I' f
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>x : M<VAR>s t : Set M<VAR>Is : SmoothManifoldWithCorners I M<VAR>I's : SmoothManifoldWithCorners I' M'<VAR>ht : t ∈ 𝓝 x<GOAL>mfderivWithin I I' f (s ∩ t) x = mfderivWithin I I' f s x
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>x : M<VAR>s : Set M<VAR>Is : SmoothManifoldWithCorners I M<VAR>I's : SmoothManifoldWithCorners I' M'<VAR>h : s ∈ 𝓝 x<GOAL>mfderivWithin I I' f s x = mfderiv I I' f x
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>x : M<VAR>s : Set M<VAR>Is : SmoothManifoldWithCorners I M<VAR>I's : SmoothManifoldWithCorners I' M'<VAR>hs : IsOpen s<VAR>hx : x ∈ s<GOAL>mfderivWithin I I' f s x = mfderiv I I' f x
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>x : M<VAR>s : Set M<VAR>Is : SmoothManifoldWithCorners I M<VAR>I's : SmoothManifoldWithCorners I' M'<VAR>hs : UniqueMDiffWithinAt I s x<VAR>h : MDifferentiableAt I I' f x<GOAL>mfderivWithin I I' f s x = mfderiv I I' f x
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>x : M<VAR>Is : SmoothManifoldWithCorners I M<VAR>I's : SmoothManifoldWithCorners I' M'<VAR>x' : M<VAR>y : M'<VAR>hx : x' ∈ (chartAt H x).source<VAR>hy : f x' ∈ (chartAt H' y).source<GOAL>MDifferentiableAt I I' f x' ↔ ContinuousAt f x' ∧ DifferentiableWithinAt 𝕜 (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm) (Set.range ↑I) (↑(extChartAt I x) x')
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>x : M<VAR>s : Set M<VAR>n : ℕ∞<VAR>hf : ContMDiffWithinAt I I' n f s x<VAR>hn : 1 ≤ n<GOAL>MDifferentiableWithinAt I I' f s x
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>x : M<VAR>n : ℕ∞<VAR>hf : ContMDiffAt I I' n f x<VAR>hn : 1 ≤ n<GOAL>MDifferentiableAt I I' f x
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>s : Set M<VAR>n : ℕ∞<VAR>hf : ContMDiffOn I I' n f s<VAR>hn : 1 ≤ n<GOAL>MDifferentiableOn I I' f s
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>n : ℕ∞<VAR>hf : ContMDiff I I' n f<VAR>hn : 1 ≤ n<GOAL>MDifferentiable I I' f
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>x : M<VAR>s : Set M<VAR>hf : SmoothWithinAt I I' f s x<GOAL>MDifferentiableWithinAt I I' f s x
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>x : M<VAR>hf : SmoothAt I I' f x<GOAL>MDifferentiableAt I I' f x
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>s : Set M<VAR>hf : SmoothOn I I' f s<GOAL>MDifferentiableOn I I' f s
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>hf : Smooth I I' f<GOAL>MDifferentiable I I' f
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>x : M<VAR>hf : Smooth I I' f<GOAL>MDifferentiableAt I I' f x
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>x : M<VAR>s : Set M<VAR>hf : Smooth I I' f<GOAL>MDifferentiableWithinAt I I' f s x
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>x : M<VAR>s : Set M<VAR>Is : SmoothManifoldWithCorners I M<VAR>I's : SmoothManifoldWithCorners I' M'<VAR>f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)<VAR>h : HasMFDerivWithinAt I I' f s x f'<GOAL>ContinuousWithinAt f s x
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>x : M<VAR>Is : SmoothManifoldWithCorners I M<VAR>I's : SmoothManifoldWithCorners I' M'<VAR>f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)<VAR>h : HasMFDerivAt I I' f x f'<GOAL>ContinuousAt f x
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>s : Set M<VAR>h : MDifferentiableOn I I' f s<GOAL>ContinuousOn f s
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>h : MDifferentiable I I' f<GOAL>Continuous f
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>s t : Set M<VAR>Is : SmoothManifoldWithCorners I M<VAR>I's : SmoothManifoldWithCorners I' M'<VAR>p : TangentBundle I M<VAR>st : s ⊆ t<VAR>hs : UniqueMDiffWithinAt I s p.proj<VAR>h : MDifferentiableWithinAt I I' f t p.proj<GOAL>tangentMapWithin I I' f s p = tangentMapWithin I I' f t p
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>Is : SmoothManifoldWithCorners I M<VAR>I's : SmoothManifoldWithCorners I' M'<GOAL>tangentMapWithin I I' f Set.univ = tangentMap I I' f
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>s : Set M<VAR>Is : SmoothManifoldWithCorners I M<VAR>I's : SmoothManifoldWithCorners I' M'<VAR>p : TangentBundle I M<VAR>hs : UniqueMDiffWithinAt I s p.proj<VAR>h : MDifferentiableAt I I' f p.proj<GOAL>tangentMapWithin I I' f s p = tangentMap I I' f p
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>s : Set M<VAR>Is : SmoothManifoldWithCorners I M<VAR>I's : SmoothManifoldWithCorners I' M'<VAR>p : TangentBundle I M<GOAL>(tangentMapWithin I I' f s p).proj = f p.proj
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>f : M → M'<VAR>Is : SmoothManifoldWithCorners I M<VAR>I's : SmoothManifoldWithCorners I' M'<VAR>p : TangentBundle I M<GOAL>(tangentMap I I' f p).proj = f p.proj
<VAR>𝕜 : Type u_1<VAR>NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>NormedAddCommGroup E<VAR>NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>TopologicalSpace M<VAR>ChartedSpace H M<VAR>E' : Type u_5<VAR>NormedAddCommGroup E'<VAR>NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>TopologicalSpace M'<VAR>ChartedSpace H' M'<VAR>E'' : Type u_8<VAR>NormedAddCommGroup E''<VAR>NormedSpace 𝕜 E''<VAR>H'' : Type u_9<VAR>TopologicalSpace H''<VAR>I'' : ModelWithCorners 𝕜 E'' H''<VAR>M'' : Type u_10<VAR>TopologicalSpace M''<VAR>ChartedSpace H'' M''<VAR>x : M<VAR>s : Set M<VAR>f : M → M'<VAR>g : M → M''<VAR>hf : MDifferentiableWithinAt I I' f s x<VAR>hg : MDifferentiableWithinAt I I'' g s x<GOAL>MDifferentiableWithinAt I (I'.prod I'') (fun x => (f x, g x)) s x