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 (x1 x2 x3) x4 = x1 x2 (x1 x3 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 H1 with
x2,
x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 x8)))),
x9,
λ x10 x11 . x11 = x1 x2 (x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 x9)))))) leaving 4 subgoals.
The subproof is completed by applying H2.
Apply unknownprop_14619fcdadc5a43502995316176da02be54150d716fe5c9727e811d162c28b04 with
x0,
x1,
x3,
x4,
x5,
x6,
x7,
x8 leaving 7 subgoals.
The subproof is completed by applying H0.
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 H7.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
set y10 to be x1 x2 (x1 (x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 x8))))) x9)
set y11 to be x2 x3 (x2 x4 (x2 x5 (x2 x6 (x2 x7 (x2 x8 (x2 x9 y10))))))
Claim L10: ∀ x12 : ι → ο . x12 y11 ⟶ x12 y10
Let x12 of type ι → ο be given.
Assume H10: x12 (x3 x4 (x3 x5 (x3 x6 (x3 x7 (x3 x8 (x3 x9 (x3 y10 y11))))))).
set y13 to be λ x13 . x12
Apply unknownprop_c4ddb10955189acaa0e73a79092d08be51c123dc1e586344a586ed01c4dfb456 with
x2,
x3,
x5,
x6,
x7,
x8,
x9,
y10,
y11,
λ x14 x15 . y13 (x3 x4 x14) (x3 x4 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 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.
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.