# 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

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

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

$\blacksquare$