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.
Let x7 of type ι be given.
Let x8 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.
Assume H9: x0 x8.
Apply unknownprop_515047c02fed97c50f69ea07f84c55a116d5435d48e441446058ba601add8797 with
x0,
x1,
x3,
x4,
x5,
x6,
x7,
x8,
λ x9 x10 . x1 x2 x10 = x1 x6 (x1 x2 (x1 x8 (x1 x7 (x1 x4 (x1 x5 x3))))) leaving 9 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.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
Apply H2 with
x3,
x8,
λ x9 x10 . x1 x2 (x1 x4 (x1 x5 (x1 x6 (x1 x7 x10)))) = x1 x6 (x1 x2 (x1 x8 (x1 x7 (x1 x4 (x1 x5 x3))))) leaving 3 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H9.
Apply unknownprop_2b9e12a8bf5ac80f2f6cc8273dcec7a77ab1ba1ca1bcf0bbd93557de134effd5 with
x0,
x1,
x2,
x4,
x5,
x6,
x7,
x8,
x3 leaving 9 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
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.
The subproof is completed by applying H9.
The subproof is completed by applying H4.