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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ιιι be given.
Let x2 of type ιιι be given.
Assume H0: ∀ x3 . prim1 x3 x0∀ x4 . prim1 x4 x0x1 x3 x4 = x2 x3 x4.
Assume H1: explicit_Group x0 x1.
Claim L2: explicit_Group x0 x2
Apply explicit_Group_repindep_imp with x0, x1, x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Claim L3: prim1 (explicit_Group_identity x0 x1) x0
Apply explicit_Group_identity_in with x0, x1.
The subproof is completed by applying H1.
Claim L4: ∀ x3 . prim1 x3 x0x1 (explicit_Group_identity x0 x1) x3 = x3
Apply explicit_Group_identity_lid with x0, x1.
The subproof is completed by applying H1.
Claim L5: prim1 (explicit_Group_identity x0 x2) x0
Apply explicit_Group_identity_in with x0, x2.
The subproof is completed by applying L2.
Claim L6: ∀ x3 . prim1 x3 x0x2 x3 (explicit_Group_identity x0 x2) = x3
Apply explicit_Group_identity_rid with x0, x2.
The subproof is completed by applying L2.
Claim L7: ∀ x3 . prim1 x3 x0x1 x3 (explicit_Group_identity x0 x2) = x3
Let x3 of type ι be given.
Assume H7: prim1 x3 x0.
Apply H0 with x3, explicit_Group_identity x0 x2, λ x4 x5 . x5 = x3 leaving 3 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying L5.
Apply L6 with x3.
The subproof is completed by applying H7.
Apply explicit_Group_identity_unique with x0, x1, explicit_Group_identity x0 x1, explicit_Group_identity x0 x2 leaving 4 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying L5.
The subproof is completed by applying L4.
The subproof is completed by applying L7.