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).
Assume H2: ∀ x2 x3 . x0 x2 ⟶ x0 x3 ⟶ x1 x2 x3 = x1 x3 x2.
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.
Let x7 of type ι be given.
Assume H3: x0 x2.
Assume H4: x0 x3.
Assume H5: x0 x4.
Assume H6: x0 x5.
Assume H7: x0 x6.
Assume H8: x0 x7.
Apply H1 with
x5,
x6,
x7,
λ x8 x9 . x1 x2 (x1 x3 (x1 x4 x9)) = x1 x6 (x1 x3 (x1 x4 (x1 x7 (x1 x2 x5)))) leaving 4 subgoals.
The subproof is completed by applying H6.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
Apply H2 with
x5,
x7,
λ x8 x9 . x1 x2 (x1 x3 (x1 x4 (x1 x6 x9))) = x1 x6 (x1 x3 (x1 x4 (x1 x7 (x1 x2 x5)))) leaving 3 subgoals.
The subproof is completed by applying H6.
The subproof is completed by applying H8.
Let x8 of type ι → ι → ο be given.
Apply unknownprop_d2139aab8690439a4d0fde81fed6f693ff4b6a70d543e77a1ee9f828670c0af5 with
x0,
x1,
x6,
x3,
x4,
x7,
x2,
x5,
λ x9 x10 . x8 x10 x9 leaving 8 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H7.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
The subproof is completed by applying H8.
The subproof is completed by applying H3.
The subproof is completed by applying H6.