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.
Let x12 of type ι be given.
Let x13 of type ι be given.
Let x14 of type ι be given.
Let x15 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.
Assume H11: x0 x12.
Assume H12: x0 x13.
Assume H13: x0 x14.
Assume H14: x0 x15.
Apply H1 with
x3,
x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 (x1 x11 (x1 x12 (x1 x13 x14))))))))),
x15,
λ x16 x17 . x17 = x1 (x2 x3 x15) (x1 (x2 x4 x15) (x1 (x2 x5 x15) (x1 (x2 x6 x15) (x1 (x2 x7 x15) (x1 (x2 x8 x15) (x1 (x2 x9 x15) (x1 (x2 x10 x15) (x1 (x2 x11 x15) (x1 (x2 x12 x15) (x1 (x2 x13 x15) (x2 x14 x15))))))))))) leaving 4 subgoals.
The subproof is completed by applying H2.
Apply unknownprop_3110297454f8d445696fa1ff6d16fb16e6639939f8e803a0cc7ed7a2132c96cf with
x0,
x1,
x4,
x5,
x6,
x7,
x8,
x9,
x10,
x11,
x12,
x13,
x14 leaving 12 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.
The subproof is completed by applying H11.
The subproof is completed by applying H12.
The subproof is completed by applying H13.
The subproof is completed by applying H14.
set y16 to be ...
set y17 to be ...
Claim L15: ∀ x18 : ι → ο . x18 y17 ⟶ x18 y16
Let x18 of type ι → ο be given.
Assume H15: x18 (x3 (x4 x5 y17) (x3 (x4 x6 y17) (x3 (x4 x7 y17) (x3 (x4 x8 y17) (x3 (x4 x9 y17) (x3 (x4 x10 y17) (x3 (x4 x11 y17) (x3 (x4 x12 y17) (x3 (x4 x13 y17) (x3 (x4 x14 y17) (x3 (x4 x15 y17) (x4 y16 y17)))))))))))).
set y19 to be λ x19 . ...
Apply unknownprop_a06f9e6e6ae69bf0dd8bd48452b0aec160b9654f950749e6fd5323cc4f38949a with
x2,
x3,
x4,
x6,
x7,
x8,
x9,
x10,
x11,
x12,
x13,
x14,
x15,
y16,
y17,
λ x20 x21 . y19 (x3 (x4 x5 y17) x20) (x3 (x4 x5 y17) x21) leaving 15 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.
The subproof is completed by applying H12.
The subproof is completed by applying H13.
The subproof is completed by applying H14.
The subproof is completed by applying H15.
Let x18 of type ι → ι → ο be given.
Apply L15 with
λ x19 . x18 x19 y17 ⟶ x18 y17 x19.
Assume H16: x18 y17 y17.
The subproof is completed by applying H16.