-- Module: Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
n : WithTop ℕ∞
inst_6 : IsManifold I n M
e e' : OpenPartialHomeomorph M H
x : M
hn : n ≠ 0
he : Membership.mem (γ := Set (OpenPartialHomeomorph M H)) (atlas H M) e
he' : Membership.mem (γ := Set (OpenPartialHomeomorph M H)) (atlas H M) e'
hex : x ∈ e.source
hex' : x ∈ e'.source
hx : (↑(e.extend I) : M → E) x ∈ interior (e.extend I).target
φ : PartialEquiv E E := extendCoordChange.{u_1, u_2, u_4, u_3} (𝕜 := 𝕜) (I := I) e e'
hφ : ContDiffOn 𝕜 n (↑φ : E → E) φ.source
hφx' : Surjective (α := E) (β := E) (⇑(fderivWithin 𝕜 (↑φ : E → E) φ.source ((↑(e.extend I) : M → E) x)) : (a : E) → E)
inr :
  ∀
    (this :
      ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_7 : NormedAddCommGroup E]
        [inst_8 : NormedSpace 𝕜 E] {H : Type u_3} [inst_9 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H}
        {M : Type u_4} [inst_10 : TopologicalSpace M] [inst_11 : ChartedSpace H M] {n : WithTop ℕ∞} [IsManifold I n M]
        {e e' : OpenPartialHomeomorph M H} {x : M} (hn : n ≠ 0)
        (he : Membership.mem (γ := Set (OpenPartialHomeomorph M H)) (atlas H M) e)
        (he' : Membership.mem (γ := Set (OpenPartialHomeomorph M H)) (atlas H M) e') (hex : x ∈ e.source)
        (hex' : x ∈ e'.source) (hx : (↑(e.extend I) : M → E) x ∈ interior (e.extend I).target),
        let φ : PartialEquiv E E := extendCoordChange.{u_1, u_2, u_4, u_3} (𝕜 := 𝕜) (I := I) e e';
        ∀ (hφ : ContDiffOn 𝕜 n (↑φ : E → E) φ.source)
          (hφx' :
            Surjective (α := E) (β := E)
              (⇑(fderivWithin 𝕜 (↑φ : E → E) φ.source ((↑(e.extend I) : M → E) x)) : (a : E) → E))
          (h : IsRCLikeNormedField 𝕜), (↑(e'.extend I) : M → E) x ∈ interior (range (↑I : H → E))),
    ¬IsRCLikeNormedField 𝕜 → (↑(e'.extend I) : M → E) x ∈ interior (range (↑I : H → E))
wlog :
  ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_7 : NormedAddCommGroup E]
    [inst_8 : NormedSpace 𝕜 E] {H : Type u_3} [inst_9 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4}
    [inst_10 : TopologicalSpace M] [inst_11 : ChartedSpace H M] {n : WithTop ℕ∞} [IsManifold I n M]
    {e e' : OpenPartialHomeomorph M H} {x : M} (hn : n ≠ 0)
    (he : Membership.mem (γ := Set (OpenPartialHomeomorph M H)) (atlas H M) e)
    (he' : Membership.mem (γ := Set (OpenPartialHomeomorph M H)) (atlas H M) e') (hex : x ∈ e.source)
    (hex' : x ∈ e'.source) (hx : (↑(e.extend I) : M → E) x ∈ interior (e.extend I).target),
    let φ : PartialEquiv E E := extendCoordChange.{u_1, u_2, u_4, u_3} (𝕜 := 𝕜) (I := I) e e';
    ∀ (hφ : ContDiffOn 𝕜 n (↑φ : E → E) φ.source)
      (hφx' :
        Surjective (α := E) (β := E)
          (⇑(fderivWithin 𝕜 (↑φ : E → E) φ.source ((↑(e.extend I) : M → E) x)) : (a : E) → E)),
      IsRCLikeNormedField 𝕜 → (↑(e'.extend I) : M → E) x ∈ interior (range (↑I : H → E))
⊢ (↑(e'.extend I) : M → E) x ∈ interior (range (↑I : H → E))

Tactic:
wlog _ : IsRCLikeNormedField 𝕜

Premises used: IsRCLikeNormedField