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).
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.
Let x9 of type ι be given.
Let x10 of type ι be given.
Let x11 of type ι be given.
Let x12 of type ι be given.
Assume H2: x0 x2.
Assume H3: x0 x3.
Assume H4: x0 x4.
Assume H5: x0 x5.
Assume H6: x0 x6.
Assume H7: x0 x7.
Assume H8: x0 x8.
Assume H9: x0 x9.
Assume H10: x0 x10.
Assume H11: x0 x11.
Assume H12: x0 x12.
Apply H1 with
x2,
x3,
x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 (x1 x11 x12))))))),
λ x13 x14 . x14 = x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 (x1 x11 (x1 x2 x12))))))))) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Apply H0 with
x4,
x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 (x1 x11 x12)))))) leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H0 with
x5,
x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 (x1 x11 x12))))) leaving 2 subgoals.
The subproof is completed by applying H5.
Apply H0 with
x6,
x1 x7 (x1 x8 (x1 x9 (x1 x10 (x1 x11 x12)))) leaving 2 subgoals.
The subproof is completed by applying H6.
Apply H0 with
x7,
x1 x8 (x1 x9 (x1 x10 (x1 x11 x12))) leaving 2 subgoals.
The subproof is completed by applying H7.
Apply H0 with
x8,
x1 x9 (x1 x10 (x1 x11 x12)) leaving 2 subgoals.
The subproof is completed by applying H8.
Apply H0 with
x9,
x1 x10 (x1 x11 x12) leaving 2 subgoals.
The subproof is completed by applying H9.
Apply H0 with
x10,
x1 x11 x12 leaving 2 subgoals.
The subproof is completed by applying H10.
Apply H0 with
x11,
x12 leaving 2 subgoals.
The subproof is completed by applying H11.
The subproof is completed by applying H12.
set y13 to be ...
set y14 to be x2 x4 (x2 x5 (x2 x6 (x2 x7 ...)))
Claim L13: ∀ x15 : ι → ο . x15 y14 ⟶ x15 y13
Let x15 of type ι → ο be given.
Assume H13: x15 (x3 x5 (x3 x6 (x3 x7 (x3 x8 (x3 x9 (x3 x10 (x3 x11 (x3 x12 (x3 y13 (x3 x4 y14)))))))))).
set y16 to be λ x16 . x15
Apply unknownprop_57b72403a262e58313ce568ea812ef6d89ec67b4e1a215ab6f60d964bf3d2756 with
x2,
x3,
x4,
x6,
x7,
x8,
x9,
x10,
x11,
x12,
y13,
y14,
λ x17 x18 . y16 (x3 x5 x17) (x3 x5 x18) leaving 13 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
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.
The subproof is completed by applying H10.
The subproof is completed by applying H11.
The subproof is completed by applying H12.
The subproof is completed by applying H13.
Let x15 of type ι → ι → ο be given.
Apply L13 with
λ x16 . x15 x16 y14 ⟶ x15 y14 x16.
Assume H14: x15 y14 y14.
The subproof is completed by applying H14.