Let x0 of type ι → ο be given.
Let x1 of type ι → ι → ι be given.
Assume H0: ∀ x2 x3 . x0 x2 ⟶ x0 x3 ⟶ x0 (x1 x2 x3).
Assume H1: ∀ x2 x3 x4 . x0 x2 ⟶ x0 x3 ⟶ x0 x4 ⟶ x1 x2 (x1 x3 x4) = x1 x3 (x1 x2 x4).
Let x2 of type ι be given.
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 x2.
Assume H3: x0 x3.
Assume H4: x0 x4.
Assume H5: x0 x5.
Assume H6: x0 x6.
Apply H1 with
x2,
x3,
x1 x4 (x1 x5 x6),
λ x7 x8 . x8 = x1 x3 (x1 x2 (x1 x5 (x1 x4 x6))) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Apply H0 with
x4,
x1 x5 x6 leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H0 with
x5,
x6 leaving 2 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
set y7 to be x1 x3 (x1 x2 (x1 x4 (x1 x5 x6)))
set y8 to be x2 x4 (x2 x3 (x2 x6 (x2 x5 y7)))
Claim L7: ∀ x9 : ι → ο . x9 y8 ⟶ x9 y7
Let x9 of type ι → ο be given.
Assume H7: x9 (x3 x5 (x3 x4 (x3 y7 (x3 x6 y8)))).
set y10 to be λ x10 . x9
set y11 to be x3 x4 (x3 x6 (x3 y7 y8))
set y12 to be x4 x5 (x4 y8 (x4 y7 x9))
Claim L8: ∀ x13 : ι → ο . x13 y12 ⟶ x13 y11
Let x13 of type ι → ο be given.
Assume H8: x13 (x5 x6 (x5 x9 (x5 y8 y10))).
set y14 to be λ x14 . x13
Apply H1 with
y8,
x9,
y10,
λ x15 x16 . y14 (x5 x6 x15) (x5 x6 x16) leaving 4 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
The subproof is completed by applying H8.
set y13 to be λ x13 x14 . y12 (x5 y7 x13) (x5 y7 x14)
Apply L8 with
λ x14 . y13 x14 y12 ⟶ y13 y12 x14 leaving 2 subgoals.
Assume H9: y13 y12 y12.
The subproof is completed by applying H9.
The subproof is completed by applying L8.
Let x9 of type ι → ι → ο be given.
Apply L7 with
λ x10 . x9 x10 y8 ⟶ x9 y8 x10.
Assume H8: x9 y8 y8.
The subproof is completed by applying H8.