Restriction of Operation Distributivity

Theorem
Let $\left({S, *, \circ}\right)$ be an algebraic structure.

Let $T \subseteq S$.

If the operation $\circ$ is distributive over $*$ in $\left({S, *, \circ}\right)$, then it is also distributive over $*$ on a restriction $\left({T, * \restriction_T, \circ \restriction_T}\right)$.

Proof
Similarly for $\left({a *_T b}\right) \circ_T c = \left({a \circ_T c}\right) *_T \left({b \circ_T c}\right)$.