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 . x0 x3 ⟶ x0 x4 ⟶ x0 (x2 x3 x4).
Assume H2: ∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x1 x3 (x1 x4 x5) = x1 x4 (x1 x3 x5).
Assume H3: ∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x1 (x1 x3 x4) x5 = x1 x3 (x1 x4 x5).
Assume H4: ∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x2 x3 (x1 x4 x5) = x1 (x2 x3 x4) (x2 x3 x5).
Assume H5: ∀ 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 H6: x0 x3.
Assume H7: x0 x4.
Assume H8: x0 x5.
Assume H9: x0 x6.
Assume H10: x0 x7.
Assume H11: x0 x8.
Assume H12: x0 x9.
Assume H13: x0 x10.
Apply unknownprop_179a1535f94cd7379429e1a7ffc70833758456c0c7ade38cc575943fdb821f8f with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x1 x9 x10,
λ x11 x12 . x12 = x1 (x2 x3 x7) (x1 (x2 x3 x8) (x1 (x2 x3 x9) (x1 (x2 x3 x10) (x1 (x2 x4 x7) (x1 (x2 x4 x8) (x1 (x2 x4 x9) (x1 (x2 x4 x10) (x1 (x2 x5 x7) (x1 (x2 x5 x8) (x1 (x2 x5 x9) (x1 (x2 x5 x10) (x1 (x2 x6 x7) (x1 (x2 x6 x8) (x1 (x2 x6 x9) (x2 x6 x10))))))))))))))) leaving 14 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 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.
Apply H0 with
x9,
x10 leaving 2 subgoals.
The subproof is completed by applying H12.
The subproof is completed by applying H13.
set y11 to be ...
set y12 to be ...
Claim L14: ∀ x13 : ι → ο . x13 y12 ⟶ x13 y11
Let x13 of type ι → ο be given.
Assume H14: x13 (x3 (x4 x5 x9) (x3 (x4 x5 x10) (x3 (x4 x5 y11) (x3 (x4 x5 y12) (x3 (x4 x6 x9) (x3 (x4 x6 x10) (x3 (x4 x6 y11) (x3 ... ...)))))))).
Let x13 of type ι → ι → ο be given.
Apply L14 with
λ x14 . x13 x14 y12 ⟶ x13 y12 x14.
Assume H15: x13 y12 y12.
The subproof is completed by applying H15.