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.
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.
Apply unknownprop_00c10e341cdfb1a862eb3a06ebc6201af2d6c0a18fd638a0f92f4b8c9e4bf04e with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x1 x8 x9,
λ x10 x11 . x11 = x1 (x2 x3 x7) (x1 (x2 x3 x8) (x1 (x2 x3 x9) (x1 (x2 x4 x7) (x1 (x2 x4 x8) (x1 (x2 x4 x9) (x1 (x2 x5 x7) (x1 (x2 x5 x8) (x1 (x2 x5 x9) (x1 (x2 x6 x7) (x1 (x2 x6 x8) (x2 x6 x9))))))))))) leaving 13 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.
Apply H0 with
x8,
x9 leaving 2 subgoals.
The subproof is completed by applying H11.
The subproof is completed by applying H12.
set y10 to be ...
set y11 to be ...
Claim L13: ∀ x12 : ι → ο . x12 y11 ⟶ x12 y10
Let x12 of type ι → ο be given.
Assume H13: x12 (x3 (x4 x5 x9) (x3 (x4 x5 y10) (x3 (x4 x5 y11) (x3 (x4 x6 x9) (x3 (x4 x6 y10) (x3 (x4 x6 y11) (x3 (x4 x7 x9) (x3 (x4 x7 y10) (x3 (x4 x7 y11) (x3 (x4 x8 x9) (x3 (x4 x8 y10) (x4 x8 y11)))))))))))).
set y13 to be ...
Apply H4 with
x5,
y10,
...,
...,
... leaving 5 subgoals.
Let x12 of type ι → ι → ο be given.
Apply L13 with
λ x13 . x12 x13 y11 ⟶ x12 y11 x13.
Assume H14: x12 y11 y11.
The subproof is completed by applying H14.