Extension Theorem for Distributive Operations/Identity

Theorem
Let $\circ'$ be the unique operation on $T$ which distributes over $*$ in $T$ and induces on $R$ the operation $\circ$.

Then:
 * If $e$ is an identity for $\circ$, then $e$ is also an identity for $\circ'$.

Proof
We have that $\circ'$ exists and is unique by Extension Theorem for Distributive Operations: Existence and Uniqueness.

Let $e$ be the identity element of $\struct {R, \circ}$.

Then the restrictions to $R$ of the endomorphisms $\lambda_e: x \mapsto e \circ' x$ and $\rho_e: x \mapsto x \circ' e$ of $\struct {T, *}$ are monomorphisms.

But then $\lambda_e$ and $\rho_e$ are monomorphisms by the Extension Theorem for Homomorphisms.

Hence it follows that $e$ is the identity element of $T$.