l3lab/ntp-mathlib huggingface dataset:
/-- Given a set `t` such that `s ∩ t` is dense in `s`, then the range of `derivWithin f s` is
contained in the closure of the submodule spanned by the image of `t`. -/
theorem range_derivWithin_subset_closure_span_image
    (f : 𝕜 → F) {s t : Set 𝕜} (h : s ⊆ closure (s ∩ t)) :
    range (derivWithin f s) ⊆ closure (Submodule.span 𝕜 (f '' t)) := by
  rintro - ⟨x, rfl⟩
  rcases eq_or_neBot (𝓝[s \ {x}] x) with H|H
  · simp [derivWithin, fderivWithin, H]
    exact subset_closure (zero_mem _)
  by_cases H' : DifferentiableWithinAt 𝕜 f s x; swap
  · rw [derivWithin_zero_of_not_differentiableWithinAt H']
    exact subset_closure (zero_mem _)
  have I : (𝓝[(s ∩ t) \ {x}] x).NeBot := by
    rw [← mem_closure_iff_nhdsWithin_neBot] at H ⊢
    have A : closure (s \ {x}) ⊆ closure (closure (s ∩ t) \ {x}) :=
      closure_mono (diff_subset_diff_left h)
    have B : closure (s ∩ t) \ {x} ⊆ closure ((s ∩ t) \ {x}) := by
      convert closure_diff; 

Proof state:
case h.e'_3.h.e'_4
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f✝ f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x✝ : 𝕜
s✝ t✝ : Set 𝕜
L L₁ L₂ : Filter 𝕜
f : 𝕜 → F
s t : Set 𝕜
h : s ⊆ closure (s ∩ t)
x : 𝕜
H : x ∈ closure (s \ {x})
H' : DifferentiableWithinAt 𝕜 f s x
A : closure (s \ {x}) ⊆ closure (closure (s ∩ t) \ {x})
⊢ {x} = closure {x}

Next tactic:
exact closure_singleton.symm
/-- Given a set `t` such that `s ∩ t` is dense in `s`, then the range of `derivWithin f s` is
contained in the closure of the submodule spanned by the image of `t`. -/
theorem range_derivWithin_subset_closure_span_image
    (f : 𝕜 → F) {s t : Set 𝕜} (h : s ⊆ closure (s ∩ t)) :
    range (derivWithin f s) ⊆ closure (Submodule.span 𝕜 (f '' t)) := by
  rintro - ⟨x, rfl⟩
  rcases eq_or_neBot (𝓝[s \ {x}] x) with H|H
  · simp [derivWithin, fderivWithin, H]
    exact subset_closure (zero_mem _)
  by_cases H' : DifferentiableWithinAt 𝕜 f s x; swap
  · rw [derivWithin_zero_of_not_differentiableWithinAt H']
    exact subset_closure (zero_mem _)
  have I : (𝓝[(s ∩ t) \ {x}] x).NeBot := by
    rw [← mem_closure_iff_nhdsWithin_neBot] at H ⊢
    have A : closure (s \ {x}) ⊆ closure (closure (s ∩ t) \ {x}) :=
      closure_mono (diff_subset_diff_left h)
    have B : closure (s ∩ t) \ {x} ⊆ closure ((s ∩ t) \ {x}) := by
      convert closure_diff; exact closure_singleton.symm
    

Proof state:
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f✝ f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x✝ : 𝕜
s✝ t✝ : Set 𝕜
L L₁ L₂ : Filter 𝕜
f : 𝕜 → F
s t : Set 𝕜
h : s ⊆ closure (s ∩ t)
x : 𝕜
H : x ∈ closure (s \ {x})
H' : DifferentiableWithinAt 𝕜 f s x
A : closure (s \ {x}) ⊆ closure (closure (s ∩ t) \ {x})
B : closure (s ∩ t) \ {x} ⊆ closure ((s ∩ t) \ {x})
⊢ x ∈ closure ((s ∩ t) \ {x})

Next tactic:
simpa using A.trans (closure_mono B) H
/-- Given a set `t` such that `s ∩ t` is dense in `s`, then the range of `derivWithin f s` is
contained in the closure of the submodule spanned by the image of `t`. -/
theorem range_derivWithin_subset_closure_span_image
    (f : 𝕜 → F) {s t : Set 𝕜} (h : s ⊆ closure (s ∩ t)) :
    range (derivWithin f s) ⊆ closure (Submodule.span 𝕜 (f '' t)) := by
  rintro - ⟨x, rfl⟩
  rcases eq_or_neBot (𝓝[s \ {x}] x) with H|H
  · simp [derivWithin, fderivWithin, H]
    exact subset_closure (zero_mem _)
  by_cases H' : DifferentiableWithinAt 𝕜 f s x; swap
  · rw [derivWithin_zero_of_not_differentiableWithinAt H']
    exact subset_closure (zero_mem _)
  have I : (𝓝[(s ∩ t) \ {x}] x).NeBot := by
    rw [← mem_closure_iff_nhdsWithin_neBot] at H ⊢
    have A : closure (s \ {x}) ⊆ closure (closure (s ∩ t) \ {x}) :=
      closure_mono (diff_subset_diff_left h)
    have B : closure (s ∩ t) \ {x} ⊆ closure ((s ∩ t) \ {x}) := by
      convert closure_diff; exact closure_singleton.symm
    simpa using A.trans (closure_mono B) H
  

Proof state:
case pos
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f✝ f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x✝ : 𝕜
s✝ t✝ : Set 𝕜
L L₁ L₂ : Filter 𝕜
f : 𝕜 → F
s t : Set 𝕜
h : s ⊆ closure (s ∩ t)
x : 𝕜
H : NeBot (𝓝[s \ {x}] x)
H' : DifferentiableWithinAt 𝕜 f s x
I : NeBot (𝓝[(s ∩ t) \ {x}] x)
⊢ derivWithin f s x ∈ closure ↑(Submodule.span 𝕜 (f '' t))

Next tactic:
have : Tendsto (slope f x) (𝓝[(s ∩ t) \ {x}] x) (𝓝 (derivWithin f s x)) := by
    apply Tendsto.mono_left (hasDerivWithinAt_iff_tendsto_slope.1 H'.hasDerivWithinAt)
    rw [inter_comm, inter_diff_assoc]
    exact nhdsWithin_mono _ (inter_subset_right _ _)
/-- Given a set `t` such that `s ∩ t` is dense in `s`, then the range of `derivWithin f s` is
contained in the closure of the submodule spanned by the image of `t`. -/
theorem range_derivWithin_subset_closure_span_image
    (f : 𝕜 → F) {s t : Set 𝕜} (h : s ⊆ closure (s ∩ t)) :
    range (derivWithin f s) ⊆ closure (Submodule.span 𝕜 (f '' t)) := by
  rintro - ⟨x, rfl⟩
  rcases eq_or_neBot (𝓝[s \ {x}] x) with H|H
  · simp [derivWithin, fderivWithin, H]
    exact subset_closure (zero_mem _)
  by_cases H' : DifferentiableWithinAt 𝕜 f s x; swap
  · rw [derivWithin_zero_of_not_differentiableWithinAt H']
    exact subset_closure (zero_mem _)
  have I : (𝓝[(s ∩ t) \ {x}] x).NeBot := by
    rw [← mem_closure_iff_nhdsWithin_neBot] at H ⊢
    have A : closure (s \ {x}) ⊆ closure (closure (s ∩ t) \ {x}) :=
      closure_mono (diff_subset_diff_left h)
    have B : closure (s ∩ t) \ {x} ⊆ closure ((s ∩ t) \ {x}) := by
      convert closure_diff; exact closure_singleton.symm
    simpa using A.trans (closure_mono B) H
  have : Tendsto (slope f x) (𝓝[(s ∩ t) \ {x}] x) (𝓝 (derivWithin f s x)) := by
    

Proof state:
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f✝ f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x✝ : 𝕜
s✝ t✝ : Set 𝕜
L L₁ L₂ : Filter 𝕜
f : 𝕜 → F
s t : Set 𝕜
h : s ⊆ closure (s ∩ t)
x : 𝕜
H : NeBot (𝓝[s \ {x}] x)
H' : DifferentiableWithinAt 𝕜 f s x
I : NeBot (𝓝[(s ∩ t) \ {x}] x)
⊢ Tendsto (slope f x) (𝓝[(s ∩ t) \ {x}] x) (𝓝 (derivWithin f s x))

Next tactic:
apply Tendsto.mono_left (hasDerivWithinAt_iff_tendsto_slope.1 H'.hasDerivWithinAt)
/-- Given a set `t` such that `s ∩ t` is dense in `s`, then the range of `derivWithin f s` is
contained in the closure of the submodule spanned by the image of `t`. -/
theorem range_derivWithin_subset_closure_span_image
    (f : 𝕜 → F) {s t : Set 𝕜} (h : s ⊆ closure (s ∩ t)) :
    range (derivWithin f s) ⊆ closure (Submodule.span 𝕜 (f '' t)) := by
  rintro - ⟨x, rfl⟩
  rcases eq_or_neBot (𝓝[s \ {x}] x) with H|H
  · simp [derivWithin, fderivWithin, H]
    exact subset_closure (zero_mem _)
  by_cases H' : DifferentiableWithinAt 𝕜 f s x; swap
  · rw [derivWithin_zero_of_not_differentiableWithinAt H']
    exact subset_closure (zero_mem _)
  have I : (𝓝[(s ∩ t) \ {x}] x).NeBot := by
    rw [← mem_closure_iff_nhdsWithin_neBot] at H ⊢
    have A : closure (s \ {x}) ⊆ closure (closure (s ∩ t) \ {x}) :=
      closure_mono (diff_subset_diff_left h)
    have B : closure (s ∩ t) \ {x} ⊆ closure ((s ∩ t) \ {x}) := by
      convert closure_diff; exact closure_singleton.symm
    simpa using A.trans (closure_mono B) H
  have : Tendsto (slope f x) (𝓝[(s ∩ t) \ {x}] x) (𝓝 (derivWithin f s x)) := by
    apply Tendsto.mono_left (hasDerivWithinAt_iff_tendsto_slope.1 H'.hasDerivWithinAt)
    

Proof state:
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f✝ f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x✝ : 𝕜
s✝ t✝ : Set 𝕜
L L₁ L₂ : Filter 𝕜
f : 𝕜 → F
s t : Set 𝕜
h : s ⊆ closure (s ∩ t)
x : 𝕜
H : NeBot (𝓝[s \ {x}] x)
H' : DifferentiableWithinAt 𝕜 f s x
I : NeBot (𝓝[(s ∩ t) \ {x}] x)
⊢ 𝓝[(s ∩ t) \ {x}] x ≤ 𝓝[s \ {x}] x

Next tactic:
rw [inter_comm, inter_diff_assoc]
/-- Given a set `t` such that `s ∩ t` is dense in `s`, then the range of `derivWithin f s` is
contained in the closure of the submodule spanned by the image of `t`. -/
theorem range_derivWithin_subset_closure_span_image
    (f : 𝕜 → F) {s t : Set 𝕜} (h : s ⊆ closure (s ∩ t)) :
    range (derivWithin f s) ⊆ closure (Submodule.span 𝕜 (f '' t)) := by
  rintro - ⟨x, rfl⟩
  rcases eq_or_neBot (𝓝[s \ {x}] x) with H|H
  · simp [derivWithin, fderivWithin, H]
    exact subset_closure (zero_mem _)
  by_cases H' : DifferentiableWithinAt 𝕜 f s x; swap
  · rw [derivWithin_zero_of_not_differentiableWithinAt H']
    exact subset_closure (zero_mem _)
  have I : (𝓝[(s ∩ t) \ {x}] x).NeBot := by
    rw [← mem_closure_iff_nhdsWithin_neBot] at H ⊢
    have A : closure (s \ {x}) ⊆ closure (closure (s ∩ t) \ {x}) :=
      closure_mono (diff_subset_diff_left h)
    have B : closure (s ∩ t) \ {x} ⊆ closure ((s ∩ t) \ {x}) := by
      convert closure_diff; exact closure_singleton.symm
    simpa using A.trans (closure_mono B) H
  have : Tendsto (slope f x) (𝓝[(s ∩ t) \ {x}] x) (𝓝 (derivWithin f s x)) := by
    apply Tendsto.mono_left (hasDerivWithinAt_iff_tendsto_slope.1 H'.hasDerivWithinAt)
    rw [inter_comm, inter_diff_assoc]
    

Proof state:
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f✝ f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x✝ : 𝕜
s✝ t✝ : Set 𝕜
L L₁ L₂ : Filter 𝕜
f : 𝕜 → F
s t : Set 𝕜
h : s ⊆ closure (s ∩ t)
x : 𝕜
H : NeBot (𝓝[s \ {x}] x)
H' : DifferentiableWithinAt 𝕜 f s x
I : NeBot (𝓝[(s ∩ t) \ {x}] x)
⊢ 𝓝[t ∩ (s \ {x})] x ≤ 𝓝[s \ {x}] x

Next tactic:
exact nhdsWithin_mono _ (inter_subset_right _ _)
/-- Given a set `t` such that `s ∩ t` is dense in `s`, then the range of `derivWithin f s` is
contained in the closure of the submodule spanned by the image of `t`. -/
theorem range_derivWithin_subset_closure_span_image
    (f : 𝕜 → F) {s t : Set 𝕜} (h : s ⊆ closure (s ∩ t)) :
    range (derivWithin f s) ⊆ closure (Submodule.span 𝕜 (f '' t)) := by
  rintro - ⟨x, rfl⟩
  rcases eq_or_neBot (𝓝[s \ {x}] x) with H|H
  · simp [derivWithin, fderivWithin, H]
    exact subset_closure (zero_mem _)
  by_cases H' : DifferentiableWithinAt 𝕜 f s x; swap
  · rw [derivWithin_zero_of_not_differentiableWithinAt H']
    exact subset_closure (zero_mem _)
  have I : (𝓝[(s ∩ t) \ {x}] x).NeBot := by
    rw [← mem_closure_iff_nhdsWithin_neBot] at H ⊢
    have A : closure (s \ {x}) ⊆ closure (closure (s ∩ t) \ {x}) :=
      closure_mono (diff_subset_diff_left h)
    have B : closure (s ∩ t) \ {x} ⊆ closure ((s ∩ t) \ {x}) := by
      convert closure_diff; exact closure_singleton.symm
    simpa using A.trans (closure_mono B) H
  have : Tendsto (slope f x) (𝓝[(s ∩ t) \ {x}] x) (𝓝 (derivWithin f s x)) := by
    apply Tendsto.mono_left (hasDerivWithinAt_iff_tendsto_slope.1 H'.hasDerivWithinAt)
    rw [inter_comm, inter_diff_assoc]
    exact nhdsWithin_mono _ (inter_subset_right _ _)
  

Proof state:
case pos
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f✝ f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x✝ : 𝕜
s✝ t✝ : Set 𝕜
L L₁ L₂ : Filter 𝕜
f : 𝕜 → F
s t : Set 𝕜
h : s ⊆ closure (s ∩ t)
x : 𝕜
H : NeBot (𝓝[s \ {x}] x)
H' : DifferentiableWithinAt 𝕜 f s x
I : NeBot (𝓝[(s ∩ t) \ {x}] x)
this : Tendsto (slope f x) (𝓝[(s ∩ t) \ {x}] x) (𝓝 (derivWithin f s x))
⊢ derivWithin f s x ∈ closure ↑(Submodule.span 𝕜 (f '' t))

Next tactic:
rw [← closure_closure, ← Submodule.topologicalClosure_coe]
/-- Given a set `t` such that `s ∩ t` is dense in `s`, then the range of `derivWithin f s` is
contained in the closure of the submodule spanned by the image of `t`. -/
theorem range_derivWithin_subset_closure_span_image
    (f : 𝕜 → F) {s t : Set 𝕜} (h : s ⊆ closure (s ∩ t)) :
    range (derivWithin f s) ⊆ closure (Submodule.span 𝕜 (f '' t)) := by
  rintro - ⟨x, rfl⟩
  rcases eq_or_neBot (𝓝[s \ {x}] x) with H|H
  · simp [derivWithin, fderivWithin, H]
    exact subset_closure (zero_mem _)
  by_cases H' : DifferentiableWithinAt 𝕜 f s x; swap
  · rw [derivWithin_zero_of_not_differentiableWithinAt H']
    exact subset_closure (zero_mem _)
  have I : (𝓝[(s ∩ t) \ {x}] x).NeBot := by
    rw [← mem_closure_iff_nhdsWithin_neBot] at H ⊢
    have A : closure (s \ {x}) ⊆ closure (closure (s ∩ t) \ {x}) :=
      closure_mono (diff_subset_diff_left h)
    have B : closure (s ∩ t) \ {x} ⊆ closure ((s ∩ t) \ {x}) := by
      convert closure_diff; exact closure_singleton.symm
    simpa using A.trans (closure_mono B) H
  have : Tendsto (slope f x) (𝓝[(s ∩ t) \ {x}] x) (𝓝 (derivWithin f s x)) := by
    apply Tendsto.mono_left (hasDerivWithinAt_iff_tendsto_slope.1 H'.hasDerivWithinAt)
    rw [inter_comm, inter_diff_assoc]
    exact nhdsWithin_mono _ (inter_subset_right _ _)
  rw [← closure_closure, ← Submodule.topologicalClosure_coe]
  

Proof state:
case pos
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f✝ f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x✝ : 𝕜
s✝ t✝ : Set 𝕜
L L₁ L₂ : Filter 𝕜
f : 𝕜 → F
s t : Set 𝕜
h : s ⊆ closure (s ∩ t)
x : 𝕜
H : NeBot (𝓝[s \ {x}] x)
H' : DifferentiableWithinAt 𝕜 f s x
I : NeBot (𝓝[(s ∩ t) \ {x}] x)
this : Tendsto (slope f x) (𝓝[(s ∩ t) \ {x}] x) (𝓝 (derivWithin f s x))
⊢ derivWithin f s x ∈ closure ↑(Submodule.topologicalClosure (Submodule.span 𝕜 (f '' t)))

Next tactic:
apply mem_closure_of_tendsto this
/-- Given a set `t` such that `s ∩ t` is dense in `s`, then the range of `derivWithin f s` is
contained in the closure of the submodule spanned by the image of `t`. -/
theorem range_derivWithin_subset_closure_span_image
    (f : 𝕜 → F) {s t : Set 𝕜} (h : s ⊆ closure (s ∩ t)) :
    range (derivWithin f s) ⊆ closure (Submodule.span 𝕜 (f '' t)) := by
  rintro - ⟨x, rfl⟩
  rcases eq_or_neBot (𝓝[s \ {x}] x) with H|H
  · simp [derivWithin, fderivWithin, H]
    exact subset_closure (zero_mem _)
  by_cases H' : DifferentiableWithinAt 𝕜 f s x; swap
  · rw [derivWithin_zero_of_not_differentiableWithinAt H']
    exact subset_closure (zero_mem _)
  have I : (𝓝[(s ∩ t) \ {x}] x).NeBot := by
    rw [← mem_closure_iff_nhdsWithin_neBot] at H ⊢
    have A : closure (s \ {x}) ⊆ closure (closure (s ∩ t) \ {x}) :=
      closure_mono (diff_subset_diff_left h)
    have B : closure (s ∩ t) \ {x} ⊆ closure ((s ∩ t) \ {x}) := by
      convert closure_diff; exact closure_singleton.symm
    simpa using A.trans (closure_mono B) H
  have : Tendsto (slope f x) (𝓝[(s ∩ t) \ {x}] x) (𝓝 (derivWithin f s x)) := by
    apply Tendsto.mono_left (hasDerivWithinAt_iff_tendsto_slope.1 H'.hasDerivWithinAt)
    rw [inter_comm, inter_diff_assoc]
    exact nhdsWithin_mono _ (inter_subset_right _ _)
  rw [← closure_closure, ← Submodule.topologicalClosure_coe]
  apply mem_closure_of_tendsto this
  

Proof state:
case pos
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f✝ f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x✝ : 𝕜
s✝ t✝ : Set 𝕜
L L₁ L₂ : Filter 𝕜
f : 𝕜 → F
s t : Set 𝕜
h : s ⊆ closure (s ∩ t)
x : 𝕜
H : NeBot (𝓝[s \ {x}] x)
H' : DifferentiableWithinAt 𝕜 f s x
I : NeBot (𝓝[(s ∩ t) \ {x}] x)
this : Tendsto (slope f x) (𝓝[(s ∩ t) \ {x}] x) (𝓝 (derivWithin f s x))
⊢ ∀ᶠ (x_1 : 𝕜) in 𝓝[(s ∩ t) \ {x}] x, slope f x x_1 ∈ ↑(Submodule.topologicalClosure (Submodule.span 𝕜 (f '' t)))

Next tactic:
filter_upwards [self_mem_nhdsWithin] with y hy
/-- Given a set `t` such that `s ∩ t` is dense in `s`, then the range of `derivWithin f s` is
contained in the closure of the submodule spanned by the image of `t`. -/
theorem range_derivWithin_subset_closure_span_image
    (f : 𝕜 → F) {s t : Set 𝕜} (h : s ⊆ closure (s ∩ t)) :
    range (derivWithin f s) ⊆ closure (Submodule.span 𝕜 (f '' t)) := by
  rintro - ⟨x, rfl⟩
  rcases eq_or_neBot (𝓝[s \ {x}] x) with H|H
  · simp [derivWithin, fderivWithin, H]
    exact subset_closure (zero_mem _)
  by_cases H' : DifferentiableWithinAt 𝕜 f s x; swap
  · rw [derivWithin_zero_of_not_differentiableWithinAt H']
    exact subset_closure (zero_mem _)
  have I : (𝓝[(s ∩ t) \ {x}] x).NeBot := by
    rw [← mem_closure_iff_nhdsWithin_neBot] at H ⊢
    have A : closure (s \ {x}) ⊆ closure (closure (s ∩ t) \ {x}) :=
      closure_mono (diff_subset_diff_left h)
    have B : closure (s ∩ t) \ {x} ⊆ closure ((s ∩ t) \ {x}) := by
      convert closure_diff; exact closure_singleton.symm
    simpa using A.trans (closure_mono B) H
  have : Tendsto (slope f x) (𝓝[(s ∩ t) \ {x}] x) (𝓝 (derivWithin f s x)) := by
    apply Tendsto.mono_left (hasDerivWithinAt_iff_tendsto_slope.1 H'.hasDerivWithinAt)
    rw [inter_comm, inter_diff_assoc]
    exact nhdsWithin_mono _ (inter_subset_right _ _)
  rw [← closure_closure, ← Submodule.topologicalClosure_coe]
  apply mem_closure_of_tendsto this
  filter_upwards [self_mem_nhdsWithin] with y hy
  

Proof state:
case h
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f✝ f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x✝ : 𝕜
s✝ t✝ : Set 𝕜
L L₁ L₂ : Filter 𝕜
f : 𝕜 → F
s t : Set 𝕜
h : s ⊆ closure (s ∩ t)
x : 𝕜
H : NeBot (𝓝[s \ {x}] x)
H' : DifferentiableWithinAt 𝕜 f s x
I : NeBot (𝓝[(s ∩ t) \ {x}] x)
this : Tendsto (slope f x) (𝓝[(s ∩ t) \ {x}] x) (𝓝 (derivWithin f s x))
y : 𝕜
hy : y ∈ (s ∩ t) \ {x}
⊢ slope f x y ∈ ↑(Submodule.topologicalClosure (Submodule.span 𝕜 (f '' t)))

Next tactic:
simp only [slope, vsub_eq_sub, SetLike.mem_coe]
/-- Given a set `t` such that `s ∩ t` is dense in `s`, then the range of `derivWithin f s` is
contained in the closure of the submodule spanned by the image of `t`. -/
theorem range_derivWithin_subset_closure_span_image
    (f : 𝕜 → F) {s t : Set 𝕜} (h : s ⊆ closure (s ∩ t)) :
    range (derivWithin f s) ⊆ closure (Submodule.span 𝕜 (f '' t)) := by
  rintro - ⟨x, rfl⟩
  rcases eq_or_neBot (𝓝[s \ {x}] x) with H|H
  · simp [derivWithin, fderivWithin, H]
    exact subset_closure (zero_mem _)
  by_cases H' : DifferentiableWithinAt 𝕜 f s x; swap
  · rw [derivWithin_zero_of_not_differentiableWithinAt H']
    exact subset_closure (zero_mem _)
  have I : (𝓝[(s ∩ t) \ {x}] x).NeBot := by
    rw [← mem_closure_iff_nhdsWithin_neBot] at H ⊢
    have A : closure (s \ {x}) ⊆ closure (closure (s ∩ t) \ {x}) :=
      closure_mono (diff_subset_diff_left h)
    have B : closure (s ∩ t) \ {x} ⊆ closure ((s ∩ t) \ {x}) := by
      convert closure_diff; exact closure_singleton.symm
    simpa using A.trans (closure_mono B) H
  have : Tendsto (slope f x) (𝓝[(s ∩ t) \ {x}] x) (𝓝 (derivWithin f s x)) := by
    apply Tendsto.mono_left (hasDerivWithinAt_iff_tendsto_slope.1 H'.hasDerivWithinAt)
    rw [inter_comm, inter_diff_assoc]
    exact nhdsWithin_mono _ (inter_subset_right _ _)
  rw [← closure_closure, ← Submodule.topologicalClosure_coe]
  apply mem_closure_of_tendsto this
  filter_upwards [self_mem_nhdsWithin] with y hy
  simp only [slope, vsub_eq_sub, SetLike.mem_coe]
  

Proof state:
case h
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f✝ f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x✝ : 𝕜
s✝ t✝ : Set 𝕜
L L₁ L₂ : Filter 𝕜
f : 𝕜 → F
s t : Set 𝕜
h : s ⊆ closure (s ∩ t)
x : 𝕜
H : NeBot (𝓝[s \ {x}] x)
H' : DifferentiableWithinAt 𝕜 f s x
I : NeBot (𝓝[(s ∩ t) \ {x}] x)
this : Tendsto (slope f x) (𝓝[(s ∩ t) \ {x}] x) (𝓝 (derivWithin f s x))
y : 𝕜
hy : y ∈ (s ∩ t) \ {x}
⊢ (y - x)⁻¹ • (f y - f x) ∈ Submodule.topologicalClosure (Submodule.span 𝕜 (f '' t))

Next tactic:
refine Submodule.smul_mem _ _ (Submodule.sub_mem _ ?_ ?_)
/-- Given a set `t` such that `s ∩ t` is dense in `s`, then the range of `derivWithin f s` is
contained in the closure of the submodule spanned by the image of `t`. -/
theorem range_derivWithin_subset_closure_span_image
    (f : 𝕜 → F) {s t : Set 𝕜} (h : s ⊆ closure (s ∩ t)) :
    range (derivWithin f s) ⊆ closure (Submodule.span 𝕜 (f '' t)) := by
  rintro - ⟨x, rfl⟩
  rcases eq_or_neBot (𝓝[s \ {x}] x) with H|H
  · simp [derivWithin, fderivWithin, H]
    exact subset_closure (zero_mem _)
  by_cases H' : DifferentiableWithinAt 𝕜 f s x; swap
  · rw [derivWithin_zero_of_not_differentiableWithinAt H']
    exact subset_closure (zero_mem _)
  have I : (𝓝[(s ∩ t) \ {x}] x).NeBot := by
    rw [← mem_closure_iff_nhdsWithin_neBot] at H ⊢
    have A : closure (s \ {x}) ⊆ closure (closure (s ∩ t) \ {x}) :=
      closure_mono (diff_subset_diff_left h)
    have B : closure (s ∩ t) \ {x} ⊆ closure ((s ∩ t) \ {x}) := by
      convert closure_diff; exact closure_singleton.symm
    simpa using A.trans (closure_mono B) H
  have : Tendsto (slope f x) (𝓝[(s ∩ t) \ {x}] x) (𝓝 (derivWithin f s x)) := by
    apply Tendsto.mono_left (hasDerivWithinAt_iff_tendsto_slope.1 H'.hasDerivWithinAt)
    rw [inter_comm, inter_diff_assoc]
    exact nhdsWithin_mono _ (inter_subset_right _ _)
  rw [← closure_closure, ← Submodule.topologicalClosure_coe]
  apply mem_closure_of_tendsto this
  filter_upwards [self_mem_nhdsWithin] with y hy
  simp only [slope, vsub_eq_sub, SetLike.mem_coe]
  refine Submodule.smul_mem _ _ (Submodule.sub_mem _ ?_ ?_)
  · 

Proof state:
case h.refine_1
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f✝ f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x✝ : 𝕜
s✝ t✝ : Set 𝕜
L L₁ L₂ : Filter 𝕜
f : 𝕜 → F
s t : Set 𝕜
h : s ⊆ closure (s ∩ t)
x : 𝕜
H : NeBot (𝓝[s \ {x}] x)
H' : DifferentiableWithinAt 𝕜 f s x
I : NeBot (𝓝[(s ∩ t) \ {x}] x)
this : Tendsto (slope f x) (𝓝[(s ∩ t) \ {x}] x) (𝓝 (derivWithin f s x))
y : 𝕜
hy : y ∈ (s ∩ t) \ {x}
⊢ f y ∈ Submodule.topologicalClosure (Submodule.span 𝕜 (f '' t))

Next tactic:
apply Submodule.le_topologicalClosure
/-- Given a set `t` such that `s ∩ t` is dense in `s`, then the range of `derivWithin f s` is
contained in the closure of the submodule spanned by the image of `t`. -/
theorem range_derivWithin_subset_closure_span_image
    (f : 𝕜 → F) {s t : Set 𝕜} (h : s ⊆ closure (s ∩ t)) :
    range (derivWithin f s) ⊆ closure (Submodule.span 𝕜 (f '' t)) := by
  rintro - ⟨x, rfl⟩
  rcases eq_or_neBot (𝓝[s \ {x}] x) with H|H
  · simp [derivWithin, fderivWithin, H]
    exact subset_closure (zero_mem _)
  by_cases H' : DifferentiableWithinAt 𝕜 f s x; swap
  · rw [derivWithin_zero_of_not_differentiableWithinAt H']
    exact subset_closure (zero_mem _)
  have I : (𝓝[(s ∩ t) \ {x}] x).NeBot := by
    rw [← mem_closure_iff_nhdsWithin_neBot] at H ⊢
    have A : closure (s \ {x}) ⊆ closure (closure (s ∩ t) \ {x}) :=
      closure_mono (diff_subset_diff_left h)
    have B : closure (s ∩ t) \ {x} ⊆ closure ((s ∩ t) \ {x}) := by
      convert closure_diff; exact closure_singleton.symm
    simpa using A.trans (closure_mono B) H
  have : Tendsto (slope f x) (𝓝[(s ∩ t) \ {x}] x) (𝓝 (derivWithin f s x)) := by
    apply Tendsto.mono_left (hasDerivWithinAt_iff_tendsto_slope.1 H'.hasDerivWithinAt)
    rw [inter_comm, inter_diff_assoc]
    exact nhdsWithin_mono _ (inter_subset_right _ _)
  rw [← closure_closure, ← Submodule.topologicalClosure_coe]
  apply mem_closure_of_tendsto this
  filter_upwards [self_mem_nhdsWithin] with y hy
  simp only [slope, vsub_eq_sub, SetLike.mem_coe]
  refine Submodule.smul_mem _ _ (Submodule.sub_mem _ ?_ ?_)
  · apply Submodule.le_topologicalClosure
    

Proof state:
case h.refine_1.a
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f✝ f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x✝ : 𝕜
s✝ t✝ : Set 𝕜
L L₁ L₂ : Filter 𝕜
f : 𝕜 → F
s t : Set 𝕜
h : s ⊆ closure (s ∩ t)
x : 𝕜
H : NeBot (𝓝[s \ {x}] x)
H' : DifferentiableWithinAt 𝕜 f s x
I : NeBot (𝓝[(s ∩ t) \ {x}] x)
this : Tendsto (slope f x) (𝓝[(s ∩ t) \ {x}] x) (𝓝 (derivWithin f s x))
y : 𝕜
hy : y ∈ (s ∩ t) \ {x}
⊢ f y ∈ Submodule.span 𝕜 (f '' t)

Next tactic:
apply Submodule.subset_span