Subset Relation is Compatible with Subset Product/Corollary 2

Theorem

Let $\struct {S, \circ}$ be a magma.

Let $A, B \subseteq S$.

Let $A \subseteq B$.

Then:

\(\ds \forall x \in S: \, \)

\(\ds x \circ A\)

\(\subseteq\)

\(\ds x \circ B\)

\(\ds A \circ x\)

\(\subseteq\)

\(\ds B \circ x\)

Proof

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

$\blacksquare$