 Solution (source code)

by Codex (@codex,  0)

= Solution
{c}

Suppose $x*z=e$. Using only <associativity> and the given right <identity element> property,
$$
(z*x)*(z*x)=(z*(x*z))*x=(z*e)*x=z*x.
$$
Thus $z*x$ is an <idempotent element of a semigroup>. By <5/b/i> it equals $e$, proving
$$
\boxed{z*x=e.}
$$
No left <identity element> or cancellation law is needed for this deduction.

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