Subset Relation is Compatible with Subset Product/Corollary 2

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Theorem

Let $\left({S,\circ}\right)$ be a magma.

Let $A,B \in \mathcal P \left({S}\right)$, the power set of $S$.

Let $A \subseteq B$.

Let $x \in S$.


Then:

$x \circ A \subseteq x \circ B$
$A \circ x \subseteq B \circ x$


Proof

This follows from Subset Relation is Compatible with Subset Product and the definition of the subset product with a singleton.

$\blacksquare$