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.
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.
Apply unknownprop_01b85a5229069244bfc0fa2496398b6a876770b04992bb705aa6f0b7bd0f95bd with
x0,
x1,
x7,
x8,
x3,
x4,
x5,
x6,
x9,
λ x10 x11 . x1 x2 x10 = x1 x7 (x1 x3 (x1 x4 (x1 x6 (x1 x2 (x1 x8 (x1 x5 x9)))))) leaving 10 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 H8.
The subproof is completed by applying H3.
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 H9.
Apply H1 with
x2,
x7,
x1 x8 (x1 x3 (x1 x4 (x1 x5 (x1 x6 x9)))),
λ x10 x11 . x11 = x1 x7 (x1 x3 (x1 x4 (x1 x6 (x1 x2 (x1 x8 (x1 x5 x9)))))) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H7.
Apply H0 with
x8,
x1 x3 (x1 x4 (x1 x5 (x1 x6 x9))) leaving 2 subgoals.
The subproof is completed by applying H8.
Apply H0 with
x3,
x1 x4 (x1 x5 (x1 x6 x9)) leaving 2 subgoals.
The subproof is completed by applying H3.
Apply H0 with
x4,
x1 x5 (x1 x6 x9) leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H0 with
x5,
x1 x6 x9 leaving 2 subgoals.
The subproof is completed by applying H5.
Apply H0 with
x6,
x9 leaving 2 subgoals.
The subproof is completed by applying H6.
The subproof is completed by applying H9.
set y10 to be x1 x7 (x1 x2 (x1 x8 (x1 x3 (x1 x4 (x1 x5 (x1 x6 x9))))))
set y11 to be x2 x8 (x2 x4 (x2 x5 (x2 x7 (x2 x3 (x2 x9 (x2 x6 y10))))))
Claim L10: ∀ x12 : ι → ο . x12 y11 ⟶ x12 y10
Let x12 of type ι → ο be given.
Assume H10: x12 (x3 x9 (x3 x5 (x3 x6 (x3 x8 (x3 x4 (x3 y10 (x3 x7 y11))))))).
set y13 to be λ x13 . x12
set y14 to be λ x14 x15 . y13 (x3 x9 x14) (x3 x9 x15)
Apply unknownprop_1bd57b7201a39beba7371e10261fa6ebd3e5c1a080c5d1289172869285cc54dd with
x2,
x3,
x5,
x6,
x8,
x4,
y10,
x7,
y11,
λ x15 x16 . y14 x16 x15 leaving 10 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 H4.
The subproof is completed by applying H6.
The subproof is completed by applying H2.
The subproof is completed by applying H8.
The subproof is completed by applying H5.
The subproof is completed by applying H9.
The subproof is completed by applying H10.
Let x12 of type ι → ι → ο be given.
Apply L10 with
λ x13 . x12 x13 y11 ⟶ x12 y11 x13.
Assume H11: x12 y11 y11.
The subproof is completed by applying H11.