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