Let x0 of type ι be given.
Let 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 = x2 x3 x4.
Assume H1: ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 = x1 x4 x3.
Let x3 of type ι be given.
Assume H2: x3 ∈ x0.
Let x4 of type ι be given.
Assume H3: x4 ∈ x0.
Apply H0 with
x3,
x4,
λ x5 x6 . x5 = x2 x4 x3 leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Apply H0 with
x4,
x3,
λ x5 x6 . x1 x3 x4 = x5 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H2.
Apply H1 with
x3,
x4 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.