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 x3 (x1 x2 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.
Apply unknownprop_6df806693864a23a378ddbca02cda4bb4bc233ff1daa8914d51c06eb72ff2550 with
x0,
x1,
x3,
x4,
x5,
x6,
λ x7 x8 . x1 x2 x8 = x1 x6 (x1 x5 (x1 x4 (x1 x2 x3))) leaving 7 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
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.
Apply H2 with
x3,
x6,
λ x7 x8 . x1 x2 (x1 x4 (x1 x5 x8)) = x1 x6 (x1 x5 (x1 x4 (x1 x2 x3))) leaving 3 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H7.
Let x7 of type ι → ι → ο be given.
Apply unknownprop_d1641884db76d154e71e1df1060b33199f2bfead2411925a087979281438edfa with
x0,
x1,
x6,
x5,
x4,
x2,
x3,
λ x8 x9 . x7 x9 x8 leaving 7 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H7.
The subproof is completed by applying H6.
The subproof is completed by applying H5.
The subproof is completed by applying H3.
The subproof is completed by applying H4.