-- Module: Mathlib.Topology.Algebra.Algebra.Equiv
Goal:
R : Type u_1
A : Type u_2
B : Type u_3
C : Type u_4
inst : CommSemiring R
inst_1 : Semiring A
inst_2 : TopologicalSpace A
inst_3 : Semiring B
inst_4 : TopologicalSpace B
inst_5 : Semiring C
inst_6 : TopologicalSpace C
inst_7 : Algebra R A
inst_8 : Algebra R B
inst_9 : Algebra R C
g' : A ≃ₐ[R] B
continuous_toFun : Continuous[inst_2, inst_4] g'.toFun
continuous_invFun : Continuous[inst_4, inst_2] g'.invFun
toFun : A → B
invFun : B → A
left_inv : Function.LeftInverse invFun toFun
right_inv : Function.RightInverse invFun toFun
map_mul' :
  ∀ (x y : A),
    { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv : A ≃ B }.toFun (x * y) =
      { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv : A ≃ B }.toFun x *
        { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv : A ≃ B }.toFun y
map_add' :
  ∀ (x y : A),
    { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv : A ≃ B }.toFun (x + y) =
      { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv : A ≃ B }.toFun x +
        { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv : A ≃ B }.toFun y
commutes' :
  ∀ (r : R),
    { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv : A ≃ B }.toFun
        ((algebraMap R A : (a : R) → A) r) =
      (algebraMap R B : (a : R) → B) r
continuous_toFun_1 :
  Continuous[inst_2, inst_4]
    { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv, map_mul' := map_mul',
        map_add' := map_add', commutes' := commutes' : A ≃ₐ[R] B }.toFun
continuous_invFun_1 :
  Continuous[inst_4, inst_2]
    { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv, map_mul' := map_mul',
        map_add' := map_add', commutes' := commutes' : A ≃ₐ[R] B }.invFun
h₁ :
  { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv, map_mul' := map_mul',
        map_add' := map_add', commutes' := commutes', continuous_toFun := continuous_toFun_1,
        continuous_invFun := continuous_invFun_1 : A ≃A[R] B }.toFun =
    { toAlgEquiv := g', continuous_toFun := continuous_toFun, continuous_invFun := continuous_invFun : A ≃A[R] B }.toFun
h₂ :
  { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv, map_mul' := map_mul',
        map_add' := map_add', commutes' := commutes', continuous_toFun := continuous_toFun_1,
        continuous_invFun := continuous_invFun_1 : A ≃A[R] B }.invFun =
    { toAlgEquiv := g', continuous_toFun := continuous_toFun, continuous_invFun := continuous_invFun :
        A ≃A[R] B }.invFun
rcases :
  ∀ (toFun_1 : A → B) (invFun_1 : B → A) (left_inv_1 : Function.LeftInverse invFun_1 toFun_1)
    (right_inv_1 : Function.RightInverse invFun_1 toFun_1)
    (map_mul'_1 :
      ∀ (x y : A),
        { toFun := toFun_1, invFun := invFun_1, left_inv := left_inv_1, right_inv := right_inv_1 : A ≃ B }.toFun
            (x * y) =
          { toFun := toFun_1, invFun := invFun_1, left_inv := left_inv_1, right_inv := right_inv_1 : A ≃ B }.toFun x *
            { toFun := toFun_1, invFun := invFun_1, left_inv := left_inv_1, right_inv := right_inv_1 : A ≃ B }.toFun y)
    (map_add'_1 :
      ∀ (x y : A),
        { toFun := toFun_1, invFun := invFun_1, left_inv := left_inv_1, right_inv := right_inv_1 : A ≃ B }.toFun
            (x + y) =
          { toFun := toFun_1, invFun := invFun_1, left_inv := left_inv_1, right_inv := right_inv_1 : A ≃ B }.toFun x +
            { toFun := toFun_1, invFun := invFun_1, left_inv := left_inv_1, right_inv := right_inv_1 : A ≃ B }.toFun y)
    (commutes'_1 :
      ∀ (r : R),
        { toFun := toFun_1, invFun := invFun_1, left_inv := left_inv_1, right_inv := right_inv_1 : A ≃ B }.toFun
            ((algebraMap R A : (a : R) → A) r) =
          (algebraMap R B : (a : R) → B) r)
    (continuous_toFun :
      Continuous[inst_2, inst_4]
        { toFun := toFun_1, invFun := invFun_1, left_inv := left_inv_1, right_inv := right_inv_1,
            map_mul' := map_mul'_1, map_add' := map_add'_1, commutes' := commutes'_1 : A ≃ₐ[R] B }.toFun)
    (continuous_invFun :
      Continuous[inst_4, inst_2]
        { toFun := toFun_1, invFun := invFun_1, left_inv := left_inv_1, right_inv := right_inv_1,
            map_mul' := map_mul'_1, map_add' := map_add'_1, commutes' := commutes'_1 : A ≃ₐ[R] B }.invFun)
    (h₁ :
      { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv, map_mul' := map_mul',
            map_add' := map_add', commutes' := commutes', continuous_toFun := continuous_toFun_1,
            continuous_invFun := continuous_invFun_1 : A ≃A[R] B }.toFun =
        { toFun := toFun_1, invFun := invFun_1, left_inv := left_inv_1, right_inv := right_inv_1,
            map_mul' := map_mul'_1, map_add' := map_add'_1, commutes' := commutes'_1,
            continuous_toFun := continuous_toFun, continuous_invFun := continuous_invFun : A ≃A[R] B }.toFun)
    (h₂ :
      { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv, map_mul' := map_mul',
            map_add' := map_add', commutes' := commutes', continuous_toFun := continuous_toFun_1,
            continuous_invFun := continuous_invFun_1 : A ≃A[R] B }.invFun =
        { toFun := toFun_1, invFun := invFun_1, left_inv := left_inv_1, right_inv := right_inv_1,
            map_mul' := map_mul'_1, map_add' := map_add'_1, commutes' := commutes'_1,
            continuous_toFun := continuous_toFun, continuous_invFun := continuous_invFun : A ≃A[R] B }.invFun),
    { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv, map_mul' := map_mul',
        map_add' := map_add', commutes' := commutes', continuous_toFun := continuous_toFun_1,
        continuous_invFun := continuous_invFun_1 : A ≃A[R] B } =
      { toFun := toFun_1, invFun := invFun_1, left_inv := left_inv_1, right_inv := right_inv_1, map_mul' := map_mul'_1,
        map_add' := map_add'_1, commutes' := commutes'_1, continuous_toFun := continuous_toFun,
        continuous_invFun := continuous_invFun }
⊢ { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv, map_mul' := map_mul',
      map_add' := map_add', commutes' := commutes', continuous_toFun := continuous_toFun_1,
      continuous_invFun := continuous_invFun_1 : A ≃A[R] B } =
    { toAlgEquiv := g', continuous_toFun := continuous_toFun, continuous_invFun := continuous_invFun }

Tactic:
rcases g' with ⟨⟨_, _⟩, _⟩