Subset Relation is Compatible with Subset Product/Corollary 2
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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.
$\blacksquare$