Image of Set Difference under Relation/Corollary 1

Corollary to Image of Set Difference under Relation
Let $\RR \subseteq S \times T$ be a relation.

Let $A \subseteq B \subseteq S$.

Then:
 * $\relcomp {\RR \sqbrk B} {\RR \sqbrk A} \subseteq \RR \sqbrk {\relcomp B A}$

where:
 * $\RR \sqbrk B$ denotes the image of $B$ under $\RR$
 * $\complement$ (in this context) denotes relative complement.

Proof
We have that $A \subseteq B$.

Then by definition of relative complement:
 * $\relcomp B A = B \setminus A$
 * $\relcomp {\RR \sqbrk B} {\RR \sqbrk A} = \RR \sqbrk B \setminus \RR \sqbrk A$

Hence, when $A \subseteq B$:
 * $\relcomp {\RR \sqbrk B} {\RR \sqbrk A} \subseteq \RR \sqbrk {\relcomp B A}$

means exactly the same thing as:
 * $\RR \sqbrk B \setminus \RR \sqbrk A \subseteq \RR \sqbrk {B \setminus A}$