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.
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.
Claim L10: ∀ x9 x10 x11 . x0 x9 ⟶ x0 x10 ⟶ x0 x11 ⟶ x1 x9 (x1 x10 x11) = x1 x10 (x1 x9 x11)
Let x9 of type ι be given.
Let x10 of type ι be given.
Let x11 of type ι be given.
Assume H10: x0 x9.
Assume H11: x0 x10.
Assume H12: x0 x11.
Apply H1 with
x10,
x9,
x11,
λ x12 x13 . x1 x9 (x1 x10 x11) = x13 leaving 4 subgoals.
The subproof is completed by applying H11.
The subproof is completed by applying H10.
The subproof is completed by applying H12.
Apply H2 with
x9,
x10,
λ x12 x13 . x1 x9 (x1 x10 x11) = x1 x12 x11 leaving 3 subgoals.
The subproof is completed by applying H10.
The subproof is completed by applying H11.
Apply H1 with
x9,
x10,
x11 leaving 3 subgoals.
The subproof is completed by applying H10.
The subproof is completed by applying H11.
The subproof is completed by applying H12.
Apply unknownprop_97ffb2c2ff10ea3877929076f0dd60854d5fe11d40843376a67625a1f315107b with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8 leaving 10 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L10.
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.