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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ιι be given.
Assume H0: ∀ x3 . prim1 x3 x0∀ x4 . prim1 x4 x0x2 x3 = x2 x4x3 = x4.
Let x3 of type ι be given.
Assume H1: prim1 x3 x0.
Claim L2: and (prim1 (inv x0 x2 (x2 x3)) x0) (x2 (inv x0 x2 (x2 x3)) = x2 x3)
Apply Eps_i_ax with λ x4 . and (prim1 x4 x0) (x2 x4 = x2 x3), x3.
Apply andI with prim1 x3 x0, x2 x3 = x2 x3 leaving 2 subgoals.
The subproof is completed by applying H1.
Let x4 of type ιιο be given.
Assume H2: x4 (x2 x3) (x2 x3).
The subproof is completed by applying H2.
Apply L2 with inv x0 x2 (x2 x3) = x3.
Assume H3: prim1 (inv x0 x2 (x2 x3)) x0.
Assume H4: x2 (inv x0 x2 (x2 x3)) = x2 x3.
Apply H0 with inv x0 x2 (x2 x3), x3 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H1.
The subproof is completed by applying H4.