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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).
Assume H2: ∀ x2 x3 . x0 x2x0 x3x1 x2 x3 = x1 x3 x2.
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.
Assume H3: x0 x2.
Assume H4: x0 x3.
Assume H5: x0 x4.
Assume H6: x0 x5.
Assume H7: x0 x6.
Assume H8: x0 x7.
Assume H9: x0 x8.
Apply H2 with x7, x8, λ x9 x10 . x1 x2 (x1 x3 (x1 x4 (x1 x5 (x1 x6 x10)))) = x1 x8 (x1 x2 (x1 x6 (x1 x3 (x1 x4 (x1 x5 x7))))) leaving 3 subgoals.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
Apply unknownprop_4ce42e2af0a2a29a07e1ea1b931136604fcf2e8b81d4d22628722cfdaf22fa72 with x0, x1, x2, x3, x4, x5, x6, x8, x7 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 H9.
The subproof is completed by applying H8.