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_5c1bc29e0f1230849e4ceeb146fe0125b3ef22f856ec61fbb8b7830c727d1e83 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.