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.
Assume H3: x0 x2.
Assume H4: x0 x3.
Assume H5: x0 x4.
Claim L6: ∀ x5 x6 x7 . x0 x5 ⟶ x0 x6 ⟶ x0 x7 ⟶ x1 x5 (x1 x6 x7) = x1 x6 (x1 x5 x7)
Let x5 of type ι be given.
Let x6 of type ι be given.
Let x7 of type ι be given.
Assume H6: x0 x5.
Assume H7: x0 x6.
Assume H8: x0 x7.
Apply H1 with
x6,
x5,
x7,
λ x8 x9 . x1 x5 (x1 x6 x7) = x9 leaving 4 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H6.
The subproof is completed by applying H8.
Apply H2 with
x5,
x6,
λ x8 x9 . x1 x5 (x1 x6 x7) = x1 x8 x7 leaving 3 subgoals.
The subproof is completed by applying H6.
The subproof is completed by applying H7.
Apply H1 with
x5,
x6,
x7 leaving 3 subgoals.
The subproof is completed by applying H6.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
Apply unknownprop_ad4dab2aba224af70e5d8a095b45560e92e0ce3be9e7748e0c75e17232c74d02 with
x0,
x1,
x2,
x3,
x4 leaving 6 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L6.
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.