Let x0 of type ι be given.
Claim L0:
∀ x1 x2 : ι → ι → ι . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 ∈ x0) ⟶ (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 = x2 x3 x4) ⟶ 28b0a.. x0 x1 ⟶ 28b0a.. x0 x2Let x1 of type ι → ι → ι be given.
Let x2 of type ι → ι → ι be given.
Assume H0: ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 ∈ x0.
Assume H1: ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 = x2 x3 x4.
Assume H2: ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x1 (x1 x3 x4) x5 = x1 x3 (x1 x4 x5).
Let x3 of type ι be given.
Assume H3: x3 ∈ x0.
Let x4 of type ι be given.
Assume H4: x4 ∈ x0.
Let x5 of type ι be given.
Assume H5: x5 ∈ x0.
Apply H1 with
x3,
x4,
λ x6 x7 . x2 x6 x5 = x2 x3 (x2 x4 x5) leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
Apply H1 with
x4,
x5,
λ x6 x7 . x2 (x1 x3 x4) x5 = x2 x3 x6 leaving 3 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
Apply H1 with
x1 x3 x4,
x5,
λ x6 x7 . x6 = x2 x3 (x1 x4 x5) leaving 3 subgoals.
Apply H0 with
x3,
x4 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
Apply H1 with
x3,
x1 x4 x5,
λ x6 x7 . x1 (x1 x3 x4) x5 = x6 leaving 3 subgoals.
The subproof is completed by applying H3.
Apply H0 with
x4,
x5 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
Apply H2 with
x3,
x4,
x5 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
Let x1 of type ι → ι → ι be given.
Assume H1: ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ∈ x0.
Let x2 of type ι → ι → ι be given.
Assume H2: ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 = x2 x3 x4.
Apply prop_ext_2 with
28b0a.. x0 x2,
28b0a.. x0 x1 leaving 2 subgoals.
Apply L0 with
x2,
x1 leaving 2 subgoals.
Let x3 of type ι be given.
Assume H3: x3 ∈ x0.
Let x4 of type ι be given.
Assume H4: x4 ∈ x0.
Apply H2 with
x3,
x4,
λ x5 x6 . x5 ∈ x0 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
Apply H1 with
x3,
x4 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
Let x3 of type ι be given.
Assume H3: x3 ∈ x0.
Let x4 of type ι be given.
Assume H4: x4 ∈ x0.
Let x5 of type ι → ι → ο be given.
Apply H2 with
x3,
x4,
λ x6 x7 . x5 x7 x6 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
Apply L0 with
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.