Restriction of Operation Distributivity

Theorem
Let $\struct {S, *, \circ}$ be an algebraic structure.

Let $T \subseteq S$.

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

Proof
Similarly for $\paren {a *_T b} \circ_T c = \paren {a \circ_T c} *_T \paren {b \circ_T c}$.