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 (x1 x2 x3) 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.
Let x8 of type ι be given.
Let x9 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.
Assume H9: x0 x8.
Assume H10: x0 x9.
Claim L11: ∀ x10 x11 x12 . x0 x10 ⟶ x0 x11 ⟶ x0 x12 ⟶ x1 x10 (x1 x11 x12) = x1 x11 (x1 x10 x12)
Let x10 of type ι be given.
Let x11 of type ι be given.
Let x12 of type ι be given.
Assume H11: x0 x10.
Assume H12: x0 x11.
Assume H13: x0 x12.
Apply H1 with
x11,
x10,
x12,
λ x13 x14 . x1 x10 (x1 x11 x12) = x14 leaving 4 subgoals.
The subproof is completed by applying H12.
The subproof is completed by applying H11.
The subproof is completed by applying H13.
Apply H2 with
x10,
x11,
λ x13 x14 . x1 x10 (x1 x11 x12) = x1 x13 x12 leaving 3 subgoals.
The subproof is completed by applying H11.
The subproof is completed by applying H12.
Apply H1 with
x10,
x11,
x12 leaving 3 subgoals.
The subproof is completed by applying H11.
The subproof is completed by applying H12.
The subproof is completed by applying H13.
Apply unknownprop_42af7464f3ab35877ec8f18b89d13cff89db9cbbd04f7080f09d5399b8ad5e2f with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x9 leaving 11 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L11.
The subproof is completed by applying H2.
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.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
The subproof is completed by applying H10.