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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_9b08836e02c4ecab23ffe407c500b75674d8128928669b1aa1e6670ede61d6f8 with x0, x1, x8, x3, x4, x5, x6, x7, x9, λ x10 x11 . x1 x2 x10 = x1 x8 (x1 x7 (x1 x2 (x1 x3 (x1 x6 (x1 x4 (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 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 H7.
The subproof is completed by applying H9.
Apply H1 with x2, x8, x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 x9)))), λ x10 x11 . x11 = x1 x8 (x1 x7 (x1 x2 (x1 x3 (x1 x6 (x1 x4 (x1 x5 x9)))))) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H8.
Apply H0 with x3, x1 x4 (x1 x5 (x1 x6 (x1 x7 x9))) leaving 2 subgoals.
The subproof is completed by applying H3.
Apply H0 with x4, x1 x5 (x1 x6 (x1 x7 x9)) leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H0 with x5, x1 x6 (x1 x7 x9) leaving 2 subgoals.
The subproof is completed by applying H5.
Apply H0 with x6, x1 x7 x9 leaving 2 subgoals.
The subproof is completed by applying H6.
Apply H0 with x7, x9 leaving 2 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H9.
set y10 to be x1 x8 (x1 x2 (x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 x9))))))
set y11 to be x2 x9 (x2 x8 (x2 x3 (x2 x4 (x2 x7 (x2 x5 (x2 x6 y10))))))
Claim L10: ∀ x12 : ι → ο . x12 y11x12 y10
Let x12 of type ιο be given.
Assume H10: x12 (x3 y10 (x3 x9 (x3 x4 (x3 x5 (x3 x8 (x3 x6 (x3 x7 y11))))))).
set y13 to be λ x13 . x12
Apply unknownprop_51f579bedcefc5f4ecc1b78a4a07c2668173b89c68b146fb294b77efb2e95563 with x2, x3, x4, x5, x6, x7, x8, x9, y11, λ x14 x15 . y13 (x3 y10 x14) (x3 y10 x15) leaving 10 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 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.