Composite of Total Relations is Total

Theorem
Let $A$ be a set.

Let $\RR$ and $\SS$ be total relations on $A$.

Then their composite $\RR \circ \SS$ is also total.

Proof
Recall the definition of composition of relations:

Hence in this particular context:
 * $\RR \circ \SS := \set {\tuple {x, z} \in A \times A: \exists y \in A: \tuple {x, y} \in \SS \land \tuple {y, z} \in \RR}$

Let $x$ and $y$ in $A$ be arbitrary.

Because $\SS$ is total, either $x \mathrel \SS y$ or $y \mathrel \SS x$.

, suppose $x \mathrel \SS y$.

From Relation is Connected and Reflexive iff Total, $\RR$ is reflexive.

That is:
 * $y \mathrel \RR y$

Thus we have:
 * $\exists y \in A: \tuple {x, y} \in \SS \land \tuple {y, y} \in \RR$

and so by definition of composition of relations:
 * $\tuple {x, y} \in \RR \circ \SS$

Similarly for $y \mathrel \SS x$:
 * $\exists x \in A: \tuple {y, x} \in \SS \land \tuple {x, x} \in \RR$

giving:
 * $\tuple {y, x} \in \RR \circ \SS$

So, by definition, $\RR \circ \SS$ is total.