<VAR>𝕜 : Type u_1<VAR>inst✝²⁴ : NontriviallyNormedField 𝕜<VAR>E : Type u_2<VAR>inst✝²³ : NormedAddCommGroup E<VAR>inst✝²² : NormedSpace 𝕜 E<VAR>H : Type u_3<VAR>inst✝²¹ : TopologicalSpace H<VAR>I : ModelWithCorners 𝕜 E H<VAR>M : Type u_4<VAR>inst✝²⁰ : TopologicalSpace M<VAR>inst✝¹⁹ : ChartedSpace H M<VAR>Is : SmoothManifoldWithCorners I M<VAR>E' : Type u_5<VAR>inst✝¹⁸ : NormedAddCommGroup E'<VAR>inst✝¹⁷ : NormedSpace 𝕜 E'<VAR>H' : Type u_6<VAR>inst✝¹⁶ : TopologicalSpace H'<VAR>I' : ModelWithCorners 𝕜 E' H'<VAR>M' : Type u_7<VAR>inst✝¹⁵ : TopologicalSpace M'<VAR>inst✝¹⁴ : ChartedSpace H' M'<VAR>I's : SmoothManifoldWithCorners I' M'<VAR>F : Type u_8<VAR>inst✝¹³ : NormedAddCommGroup F<VAR>inst✝¹² : NormedSpace 𝕜 F<VAR>G : Type u_9<VAR>inst✝¹¹ : TopologicalSpace G<VAR>J : ModelWithCorners 𝕜 F G<VAR>N : Type u_10<VAR>inst✝¹⁰ : TopologicalSpace N<VAR>inst✝⁹ : ChartedSpace G N<VAR>Js : SmoothManifoldWithCorners J N<VAR>F' : Type u_11<VAR>inst✝⁸ : NormedAddCommGroup F'<VAR>inst✝⁷ : NormedSpace 𝕜 F'<VAR>G' : Type u_12<VAR>inst✝⁶ : TopologicalSpace G'<VAR>J' : ModelWithCorners 𝕜 F' G'<VAR>N' : Type u_13<VAR>inst✝⁵ : TopologicalSpace N'<VAR>inst✝⁴ : ChartedSpace G' N'<VAR>J's : SmoothManifoldWithCorners J' N'<VAR>F₁ : Type u_14<VAR>inst✝³ : NormedAddCommGroup F₁<VAR>inst✝² : NormedSpace 𝕜 F₁<VAR>F₂ : Type u_15<VAR>inst✝¹ : NormedAddCommGroup F₂<VAR>inst✝ : NormedSpace 𝕜 F₂<VAR>f✝ f₁ : M → M'<VAR>s s₁ t : Set M<VAR>x : M<VAR>m n : ℕ∞<VAR>x₀ : N<VAR>f : N → M → M'<VAR>g : N → M<VAR>hf : ContMDiffAt (J.prod I) I' n (uncurry f) (x₀, g x₀)<VAR>hg : ContMDiffAt J I m g x₀<VAR>hmn : m + 1 ≤ n<VAR>h4f✝ : ContinuousAt (fun x => f x (g x)) x₀<VAR>h4f : (fun x => f x (g x)) ⁻¹' (extChartAt I' (f x₀ (g x₀))).source ∈ 𝓝 x₀<VAR>h3f : ∀ᶠ (x' : N × M) in 𝓝 (x₀, g x₀), ContMDiffAt (J.prod I) I' (↑One.one) (uncurry f) x'<VAR>h2f : ∀ᶠ (x₂ : N) in 𝓝 x₀, ContMDiffAt I I' 1 (f x₂) (g x₂)<VAR>h2g : g ⁻¹' (extChartAt I (g x₀)).source ∈ 𝓝 x₀<VAR>this✝² : ContDiffWithinAt 𝕜 m (fun x => fderivWithin 𝕜 (↑(extChartAt I' (f x₀ (g x₀))) ∘ f (↑(extChartAt J x₀).symm x) ∘ ↑(extChartAt I (g x₀)).symm) (range ↑I) (↑(extChartAt I (g x₀)) (g (↑(extChartAt J x₀).symm x)))) (range ↑J) (↑(extChartAt J x₀) x₀)<VAR>this✝¹ : ContMDiffAt J 𝓘(𝕜, E →L[𝕜] E') m (fun x => fderivWithin 𝕜 (↑(extChartAt I' (f x₀ (g x₀))) ∘ f x ∘ ↑(extChartAt I (g x₀)).symm) (range ↑I) (↑(extChartAt I (g x₀)) (g x))) x₀<VAR>this✝ : ContMDiffAt J 𝓘(𝕜, E →L[𝕜] E') m (fun x => fderivWithin 𝕜 (↑(extChartAt I' (f x₀ (g x₀))) ∘ ↑(extChartAt I' (f x (g x))).symm ∘ writtenInExtChartAt I I' (g x) (f x) ∘ ↑(extChartAt I (g x)) ∘ ↑(extChartAt I (g x₀)).symm) (range ↑I) (↑(extChartAt I (g x₀)) (g x))) x₀<VAR>_inst✝ : (x : N) → NormedAddCommGroup (TangentSpace I (g x)) := fun x => inferInstanceAs (NormedAddCommGroup E)<VAR>_inst : (x : N) → NormedSpace 𝕜 (TangentSpace I (g x)) := fun x => inferInstanceAs (NormedSpace 𝕜 E)<VAR>this : ContMDiffAt J 𝓘(𝕜, E →L[𝕜] E') m (fun x => (fderivWithin 𝕜 (↑(extChartAt I' (f x₀ (g x₀))) ∘ ↑(extChartAt I' (f x (g x))).symm) (range ↑I') (↑(extChartAt I' (f x (g x))) (f x (g x)))).comp ((mfderiv I I' (f x) (g x)).comp (fderivWithin 𝕜 (↑(extChartAt I (g x)) ∘ ↑(extChartAt I (g x₀)).symm) (range ↑I) (↑(extChartAt I (g x₀)) (g x))))) x₀<GOAL>inTangentCoordinates I I' g (fun x => f x (g x)) (fun x => mfderiv I I' (f x) (g x)) x₀ =ᶠ[𝓝 x₀] fun x => (fderivWithin 𝕜 (↑(extChartAt I' (f x₀ (g x₀))) ∘ ↑(extChartAt I' (f x (g x))).symm) (range ↑I') (↑(extChartAt I' (f x (g x))) (f x (g x)))).comp ((mfderiv I I' (f x) (g x)).comp (fderivWithin 𝕜 (↑(extChartAt I (g x)) ∘ ↑(extChartAt I (g x₀)).symm) (range ↑I) (↑(extChartAt I (g x₀)) (g x))))