Let x0 of type ι → ο be given.
Let x1 of type ι → ι → ι be given.
Let x2 of type ι → ι → ι be given.
Assume H0: ∀ x3 x4 . x0 x3 ⟶ x0 x4 ⟶ x0 (x1 x3 x4).
Assume H1: ∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x2 (x1 x3 x4) x5 = x1 (x2 x3 x5) (x2 x4 x5).
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.
Assume H2: x0 x3.
Assume H3: x0 x4.
Assume H4: x0 x5.
Assume H5: x0 x6.
Assume H6: x0 x7.
Assume H7: x0 x8.
Assume H8: x0 x9.
Assume H9: x0 x10.
Assume H10: x0 x11.
Apply H1 with
x3,
x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 x10))))),
x11,
λ x12 x13 . x13 = x1 (x2 x3 x11) (x1 (x2 x4 x11) (x1 (x2 x5 x11) (x1 (x2 x6 x11) (x1 (x2 x7 x11) (x1 (x2 x8 x11) (x1 (x2 x9 x11) (x2 x10 x11))))))) leaving 4 subgoals.
The subproof is completed by applying H2.
Apply unknownprop_025d233877239fdf8667e3ba4d630729f1334dc236b8bf7cefec04c2fd303300 with
x0,
x1,
x4,
x5,
x6,
x7,
x8,
x9,
x10 leaving 8 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.
The subproof is completed by applying H10.
set y12 to be x1 (x2 x3 x11) (x2 (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 x10)))))) x11)
set y13 to be x2 (x3 x4 y12) (x2 (x3 x5 y12) (x2 (x3 x6 y12) (x2 (x3 x7 y12) (x2 (x3 x8 y12) (x2 (x3 x9 y12) (x2 (x3 x10 y12) (x3 x11 y12)))))))
Claim L11: ∀ x14 : ι → ο . x14 y13 ⟶ x14 y12
Let x14 of type ι → ο be given.
Assume H11: x14 (x3 (x4 x5 y13) (x3 (x4 x6 y13) (x3 (x4 x7 y13) (x3 (x4 x8 y13) (x3 (x4 x9 y13) (x3 (x4 x10 y13) (x3 (x4 x11 y13) (x4 y12 y13)))))))).
set y15 to be λ x15 . x14
Apply unknownprop_b7b295f38ec44b7457473010f3621695d26e4e9422bae5a083cc6f30b9abc04b with
x2,
x3,
x4,
x6,
x7,
x8,
x9,
x10,
x11,
y12,
y13,
λ x16 x17 . y15 (x3 (x4 x5 y13) x16) (x3 (x4 x5 y13) x17) leaving 11 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.
The subproof is completed by applying H11.
Let x14 of type ι → ι → ο be given.
Apply L11 with
λ x15 . x14 x15 y13 ⟶ x14 y13 x15.
Assume H12: x14 y13 y13.
The subproof is completed by applying H12.