Right Inverse for All is Left Inverse

See Right Inverse for All is Left Inverse on proofwiki.org.

First we need the semigroup axioms (plus a right identity).

axiom (#id-right)  mul(A, e) = A
axiom (#assoc)     mul(A, mul(B, C)) = mul(mul(A, B), C)
axiom (#inv-right) mul(A, inv(A)) = e

Next we need Product of Semigroup Element with Right Inverse is Idempotent as a lemma.

theorem (#product-of-semigroup-element-with-right-inverse-is-idempotent)
    mul(mul(inv(A), A), mul(inv(A), A)) = mul(inv(A), A)
proof
    mul(inv(A), A) = mul(inv(A), A)
    mul(mul(inv(A), e), A) = mul(inv(A), A)
    mul(mul(inv(A), mul(A, inv(A))), A) = mul(inv(A), A)
    mul(mul(mul(inv(A), A), inv(A)), A) = mul(inv(A), A)
    mul(mul(inv(A), A), mul(inv(A), A)) = mul(inv(A), A)
qed

Finally we need this proof.

theorem (#right-inverse-for-semigroup-is-left-inverse)
    mul(inv(A), A) = e
proof
    e = e
    mul(A, inv(A)) = e
    mul(mul(inv(A), A), inv(mul(inv(A), A))) = e
                                                    [by substitution of mul(inv(A), A) into A]
    mul(mul(mul(inv(A), A), mul(inv(A), A)), inv(mul(inv(A), A))) = e
                                                    [by #product-of-semigroup-element-with-right-inverse-is-idempotent]
    mul(mul(inv(A), A), mul(mul(inv(A), A), inv(mul(inv(A), A)))) = e
    mul(mul(inv(A), A), e) = e
    mul(inv(A), A) = e
qed