Extension Theorem for Distributive Operations/Cancellability

Theorem
Then:
 * Every element of $R$ cancellable for $\circ$ is also cancellable for $\circ'$.

Proof
Let $a$ be an element of $R$ cancellable for $\circ$.

Then the restrictions to $R$ of the endomorphisms:
 * $\lambda_a: x \mapsto a \circ' x$
 * $\rho_a: x \mapsto x \circ' a$

of $\struct {T, *}$ are monomorphisms.

But then $\lambda_a$ and $\rho_a$ are monomorphisms by the Extension Theorem for Homomorphisms.

Hence $a$ is cancellable for $\circ'$.