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.
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.
Apply H1 with
x3,
x1 x4 (x1 x5 (x1 x6 (x1 x7 x8))),
x9,
λ x10 x11 . x11 = x1 (x2 x3 x9) (x1 (x2 x4 x9) (x1 (x2 x5 x9) (x1 (x2 x6 x9) (x1 (x2 x7 x9) (x2 x8 x9))))) leaving 4 subgoals.
The subproof is completed by applying H2.
Apply unknownprop_d7ce6357a8261c6a4be44f579bedcb1c2d65cec14964ea078af8f02cc5aab85a with
x0,
x1,
x4,
x5,
x6,
x7,
x8 leaving 6 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.
set y10 to be x1 (x2 x3 x9) (x2 (x1 x4 (x1 x5 (x1 x6 (x1 x7 x8)))) x9)
set y11 to be x2 (x3 x4 y10) (x2 (x3 x5 y10) (x2 (x3 x6 y10) (x2 (x3 x7 y10) (x2 (x3 x8 y10) (x3 x9 y10)))))
Claim L9: ∀ x12 : ι → ο . x12 y11 ⟶ x12 y10
Let x12 of type ι → ο be given.
Assume H9: x12 (x3 (x4 x5 y11) (x3 (x4 x6 y11) (x3 (x4 x7 y11) (x3 (x4 x8 y11) (x3 (x4 x9 y11) (x4 y10 y11)))))).
set y13 to be λ x13 . x12
Apply unknownprop_55de5c79fadd89ca3e161a61e8ef1cc68aeee5eba6c4fec4d11d6eacbce11bf5 with
x2,
x3,
x4,
x6,
x7,
x8,
x9,
y10,
y11,
λ x14 x15 . y13 (x3 (x4 x5 y11) x14) (x3 (x4 x5 y11) x15) leaving 9 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.
Let x12 of type ι → ι → ο be given.
Apply L9 with
λ x13 . x12 x13 y11 ⟶ x12 y11 x13.
Assume H10: x12 y11 y11.
The subproof is completed by applying H10.