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Proofgold Proof

pf
Let x0 of type ιο be given.
Let x1 of type ιιι be given.
Assume H0: ∀ x2 x3 . x0 x2x0 x3x0 (x1 x2 x3).
Assume H1: ∀ x2 x3 x4 . x0 x2x0 x3x0 x4x1 x2 (x1 x3 x4) = x1 x3 (x1 x2 x4).
Let x2 of type ι be given.
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 x2.
Assume H3: x0 x3.
Assume H4: x0 x4.
Assume H5: x0 x5.
Assume H6: x0 x6.
Assume H7: x0 x7.
Assume H8: x0 x8.
Assume H9: x0 x9.
Apply unknownprop_01b85a5229069244bfc0fa2496398b6a876770b04992bb705aa6f0b7bd0f95bd with x0, x1, x7, x8, x3, x4, x5, x6, x9, λ x10 x11 . x1 x2 x10 = x1 x7 (x1 x8 (x1 x3 (x1 x2 (x1 x4 (x1 x6 (x1 x5 x9)))))) leaving 10 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
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 H9.
Apply H1 with x2, x7, x1 x8 (x1 x3 (x1 x4 (x1 x5 (x1 x6 x9)))), λ x10 x11 . x11 = x1 x7 (x1 x8 (x1 x3 (x1 x2 (x1 x4 (x1 x6 (x1 x5 x9)))))) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H7.
Apply H0 with x8, x1 x3 (x1 x4 (x1 x5 (x1 x6 x9))) leaving 2 subgoals.
The subproof is completed by applying H8.
Apply H0 with x3, x1 x4 (x1 x5 (x1 x6 x9)) leaving 2 subgoals.
The subproof is completed by applying H3.
Apply H0 with x4, x1 x5 (x1 x6 x9) leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H0 with x5, x1 x6 x9 leaving 2 subgoals.
The subproof is completed by applying H5.
Apply H0 with x6, x9 leaving 2 subgoals.
The subproof is completed by applying H6.
The subproof is completed by applying H9.
set y10 to be x1 x7 (x1 x2 (x1 x8 (x1 x3 (x1 x4 (x1 x5 (x1 x6 x9))))))
set y11 to be x2 x8 (x2 x9 (x2 x4 (x2 x3 (x2 x5 (x2 x7 (x2 x6 y10))))))
Claim L10: ∀ x12 : ι → ο . x12 y11x12 y10
Let x12 of type ιο be given.
Assume H10: x12 (x3 x9 (x3 y10 (x3 x5 (x3 x4 (x3 x6 (x3 x8 (x3 x7 y11))))))).
set y13 to be λ x13 . x12
set y14 to be λ x14 x15 . y13 (x3 x9 x14) (x3 x9 x15)
Apply unknownprop_8217de59f44ad11e1e0aab82fb795d97c5f7701d63eb5d6a8602b4c147553a62 with x2, x3, y10, x5, x4, x6, x8, x7, y11, λ x15 x16 . y14 x16 x15 leaving 10 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H8.
The subproof is completed by applying H3.
The subproof is completed by applying H2.
The subproof is completed by applying H4.
The subproof is completed by applying H6.
The subproof is completed by applying H5.
The subproof is completed by applying H9.
The subproof is completed by applying H10.
Let x12 of type ιιο be given.
Apply L10 with λ x13 . x12 x13 y11x12 y11 x13.
Assume H11: x12 y11 y11.
The subproof is completed by applying H11.