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