Let x0 of type ι → ο be given.
Let x1 of type ι → ι → ι be given.
Let x2 of type ι → ι → ι be given.
Assume H0: ∀ x3 x4 . x0 x3 ⟶ x0 x4 ⟶ x0 (x1 x3 x4).
Assume H1: ∀ x3 x4 . x0 x3 ⟶ x0 x4 ⟶ x0 (x2 x3 x4).
Assume H2: ∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x1 x3 (x1 x4 x5) = x1 x4 (x1 x3 x5).
Assume H3: ∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x1 (x1 x3 x4) x5 = x1 x3 (x1 x4 x5).
Assume H4: ∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x2 x3 (x1 x4 x5) = x1 (x2 x3 x4) (x2 x3 x5).
Assume H5: ∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x2 (x1 x3 x4) x5 = x1 (x2 x3 x5) (x2 x4 x5).
Let x3 of type ι → ι be given.
Assume H6: ∀ x4 . x0 x4 ⟶ x3 x4 = x2 x4 x4.
Let x4 of type ι be given.
Let x5 of type ι be given.
Let x6 of type ι be given.
Let x7 of type ι be given.
Assume H7: x0 x4.
Assume H8: x0 x5.
Assume H9: x0 x6.
Assume H10: x0 x7.
Apply H6 with
x1 x4 (x1 x5 (x1 x6 x7)),
λ x8 x9 . x9 = x1 (x3 x4) (x1 (x2 x4 x5) (x1 (x2 x4 x6) (x1 (x2 x4 x7) (x1 (x2 x5 x4) (x1 (x3 x5) (x1 (x2 x5 x6) (x1 (x2 x5 x7) (x1 (x2 x6 x4) (x1 (x2 x6 x5) (x1 (x3 x6) (x1 (x2 x6 x7) (x1 (x2 x7 x4) (x1 (x2 x7 x5) (x1 (x2 x7 x6) (x3 x7))))))))))))))) leaving 2 subgoals.
Apply H0 with
x4,
x1 x5 (x1 x6 x7) leaving 2 subgoals.
The subproof is completed by applying H7.
Apply H0 with
x5,
x1 x6 x7 leaving 2 subgoals.
The subproof is completed by applying H8.
Apply H0 with
x6,
x7 leaving 2 subgoals.
The subproof is completed by applying H9.
The subproof is completed by applying H10.
Apply H6 with
x4,
λ x8 x9 . x2 (x1 x4 (x1 x5 (x1 x6 x7))) (x1 x4 (x1 x5 (x1 x6 x7))) = x1 x9 (x1 (x2 x4 x5) (x1 (x2 x4 x6) (x1 (x2 x4 x7) (x1 (x2 x5 x4) (x1 (x3 x5) (x1 (x2 x5 x6) (x1 (x2 x5 x7) (x1 (x2 x6 x4) (x1 (x2 ... ...) ...))))))))) leaving 2 subgoals.