Restriction of Operation Distributivity

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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 $\struct {T, * \restriction_T, \circ \restriction_T}$.


\(\ds \) \(\) \(\ds T \subseteq S\)
\(\ds \leadsto \ \ \) \(\ds \) \(\) \(\ds \forall a, b, c \in T: a, b, c \in S\) Definition of Subset
\(\ds \leadsto \ \ \) \(\ds \) \(\) \(\ds a \circ_T \paren {b *_T c}\)
\(\ds \) \(=\) \(\ds a \circ \paren {b * c}\)
\(\ds \) \(=\) \(\ds \paren {a \circ b} * \paren {a \circ c}\) as $\circ$ is distributive over $*$
\(\ds \) \(=\) \(\ds \paren {a \circ_T b} *_T \paren {a \circ_T c}\)

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