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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 (x1 x2 x3) x4.
Assume H2: ∀ x2 x3 . x0 x2x0 x3x1 x2 x3 = x1 x3 x2.
Claim L3: ∀ 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.
Assume H3: x0 x2.
Assume H4: x0 x3.
Assume H5: x0 x4.
Apply H1 with x3, x2, x4, λ x5 x6 . x1 x2 (x1 x3 x4) = x6 leaving 4 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H3.
The subproof is completed by applying H5.
Apply H2 with x2, x3, λ x5 x6 . x1 x2 (x1 x3 x4) = x1 x5 x4 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
Apply H1 with x2, x3, x4 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
Apply unknownprop_68f58642b2964b18cd9b555cf90b8361160016d975412d6db2032edaa62fef5a with x0, x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L3.