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.
Assume H3: x0 x2.
Assume H4: x0 x3.
Assume H5: x0 x4.
Assume H6: x0 x5.
Assume H7: x0 x6.
Assume H8: x0 x7.
Claim L9: ∀ x8 x9 x10 . x0 x8 ⟶ x0 x9 ⟶ x0 x10 ⟶ x1 x8 (x1 x9 x10) = x1 x9 (x1 x8 x10)
Let x8 of type ι be given.
Let x9 of type ι be given.
Let x10 of type ι be given.
Assume H9: x0 x8.
Assume H10: x0 x9.
Assume H11: x0 x10.
Apply H1 with
x9,
x8,
x10,
λ x11 x12 . x1 x8 (x1 x9 x10) = x12 leaving 4 subgoals.
The subproof is completed by applying H10.
The subproof is completed by applying H9.
The subproof is completed by applying H11.
Apply H2 with
x8,
x9,
λ x11 x12 . x1 x8 (x1 x9 x10) = x1 x11 x10 leaving 3 subgoals.
The subproof is completed by applying H9.
The subproof is completed by applying H10.
Apply H1 with
x8,
x9,
x10 leaving 3 subgoals.
The subproof is completed by applying H9.
The subproof is completed by applying H10.
The subproof is completed by applying H11.
Apply unknownprop_92e882ccf713d4f88ea99421a6e9680f28cf16aed50f757a8211b3720aa8bc60 with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7 leaving 9 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L9.
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.