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 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x2 (x1 x3 x4) x5 = x1 (x2 x3 x5) (x2 x4 x5).
Let x3 of type ι be given.
Let x4 of type ι be given.
Let x5 of type ι be given.
Let x6 of type ι be given.
Assume H2: x0 x3.
Assume H3: x0 x4.
Assume H4: x0 x5.
Assume H5: x0 x6.
Apply H1 with
x3,
x1 x4 x5,
x6,
λ x7 x8 . x8 = x1 (x2 x3 x6) (x1 (x2 x4 x6) (x2 x5 x6)) leaving 4 subgoals.
The subproof is completed by applying H2.
Apply H0 with
x4,
x5 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
set y7 to be x1 (x2 x3 x6) (x2 (x1 x4 x5) x6)
set y8 to be x2 (x3 x4 y7) (x2 (x3 x5 y7) (x3 x6 y7))
Claim L6: ∀ x9 : ι → ο . x9 y8 ⟶ x9 y7
Let x9 of type ι → ο be given.
Assume H6: x9 (x3 (x4 x5 y8) (x3 (x4 x6 y8) (x4 y7 y8))).
set y10 to be λ x10 . x9
Apply H1 with
x6,
y7,
y8,
λ x11 x12 . y10 (x3 (x4 x5 y8) x11) (x3 (x4 x5 y8) x12) leaving 4 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
Let x9 of type ι → ι → ο be given.
Apply L6 with
λ x10 . x9 x10 y8 ⟶ x9 y8 x10.
Assume H7: x9 y8 y8.
The subproof is completed by applying H7.