Extension Theorem for Distributive Operations/Commutativity

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

Then:
 * If $\circ$ is commutative, then so is $\circ'$

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

Suppose $\circ$ is commutative.

As $\circ'$ distributes over $*$, for all $n \in R$, the mappings:

are endomorphisms of $\struct {T, *}$ that coincide on $R$ by the commutativity of $\circ$ and hence are the same mapping.

Therefore $\forall x \in T, n \in R: x \circ' n = n \circ' x$.

Finally, for all $y \in T$, the mappings:

are endomorphisms of $\struct {T, *}$ that coincide on $R$ by what we have proved and hence are the same mapping.

Therefore $\circ'$ is commutative.