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.
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.
Apply H1 with
x3,
x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 x9)))),
x10,
λ x11 x12 . x12 = x1 (x2 x3 x10) (x1 (x2 x4 x10) (x1 (x2 x5 x10) (x1 (x2 x6 x10) (x1 (x2 x7 x10) (x1 (x2 x8 x10) (x2 x9 x10)))))) leaving 4 subgoals.
The subproof is completed by applying H2.
Apply unknownprop_14619fcdadc5a43502995316176da02be54150d716fe5c9727e811d162c28b04 with
x0,
x1,
x4,
x5,
x6,
x7,
x8,
x9 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 y11 to be x1 (x2 x3 x10) (x2 (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 x9))))) x10)
set y12 to be x2 (x3 x4 y11) (x2 (x3 x5 y11) (x2 (x3 x6 y11) (x2 (x3 x7 y11) (x2 (x3 x8 y11) (x2 (x3 x9 y11) (x3 x10 y11))))))
Claim L10: ∀ x13 : ι → ο . x13 y12 ⟶ x13 y11
Let x13 of type ι → ο be given.
Assume H10: x13 (x3 (x4 x5 y12) (x3 (x4 x6 y12) (x3 (x4 x7 y12) (x3 (x4 x8 y12) (x3 (x4 x9 y12) (x3 (x4 x10 y12) (x4 y11 y12))))))).
set y14 to be λ x14 . x13
Apply unknownprop_bfc1cea51a42a4a32469869e8448699c1d59a9077d0b467664f278e66191db35 with
x2,
x3,
x4,
x6,
x7,
x8,
x9,
x10,
y11,
y12,
λ x15 x16 . y14 (x3 (x4 x5 y12) x15) (x3 (x4 x5 y12) x16) 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 x13 of type ι → ι → ο be given.
Apply L10 with
λ x14 . x13 x14 y12 ⟶ x13 y12 x14.
Assume H11: x13 y12 y12.
The subproof is completed by applying H11.