Subset Relation is Compatible with Subset Product/Corollary 1
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Theorem
Let $\struct {S, \circ}$ be a magma.
Let $\struct {\powerset S, \circ_\PP}$ be the power structure of $\struct {S, \circ}$.
Let $A, B, C, D \in \powerset S$.
Let $A \subseteq B$ and $C \subseteq D$.
Then:
- $A \circ_\PP C \subseteq B \circ_\PP D$
Proof
By Subset Relation is Compatible with Subset Product, $\subseteq$ is compatible with $\circ_\PP$.
By Subset Relation is Transitive, $\subseteq$ is transitive.
Thus the theorem holds by Operating on Transitive Relationships Compatible with Operation.
$\blacksquare$