Let x0 of type ι → ο be given.
Let x1 of type (ι → ι) → ο be given.
Assume H0: ∀ x2 : ι → ι . x1 x2 ⟶ ∀ x3 . x0 x3 ⟶ x0 (x2 x3).
Assume H1: ∀ x2 x3 : ι → ι . x1 x2 ⟶ x1 x3 ⟶ ∀ x4 . x0 x4 ⟶ x2 (x3 x4) = x3 (x2 x4).
Let x2 of type ι → ι be given.
Let x3 of type ι → ι be given.
Let x4 of type ι → ι be given.
Assume H2: x1 x2.
Assume H3: x1 x3.
Assume H4: x1 x4.
Let x5 of type ι be given.
Assume H5: x0 x5.
Apply H1 with
x2,
x3,
x4 x5 leaving 3 subgoals.
The subproof is completed by applying H2.
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.