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.
Assume H3: x0 x2.
Assume H4: x0 x3.
Assume H5: x0 x4.
Assume H6: x0 x5.
Assume H7: x0 x6.
Claim L8: ∀ x7 x8 x9 . x0 x7 ⟶ x0 x8 ⟶ x0 x9 ⟶ x1 x7 (x1 x8 x9) = x1 x8 (x1 x7 x9)
Let x7 of type ι be given.
Let x8 of type ι be given.
Let x9 of type ι be given.
Assume H8: x0 x7.
Assume H9: x0 x8.
Assume H10: x0 x9.
Apply H1 with
x8,
x7,
x9,
λ x10 x11 . x1 x7 (x1 x8 x9) = x11 leaving 4 subgoals.
The subproof is completed by applying H9.
The subproof is completed by applying H8.
The subproof is completed by applying H10.
Apply H2 with
x7,
x8,
λ x10 x11 . x1 x7 (x1 x8 x9) = x1 x10 x9 leaving 3 subgoals.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
Apply H1 with
x7,
x8,
x9 leaving 3 subgoals.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
The subproof is completed by applying H10.
Apply unknownprop_d4990d6a83de07715843760ff14af7e62400cf28aa909305ee9b3bcfe536a28a with
x0,
x1,
x2,
x3,
x4,
x5,
x6 leaving 8 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L8.
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.