= Solution
If $T$ is <transitive>, then $T\circ T\subseteq T$. Since $S,R\subseteq T$, monotonicity of <composition of relations> gives
$$
S\circ R\subseteq T\circ T\subseteq T,
$$
which is condition (iii).
Conversely, condition (iii), associativity of <composition of relations>, and the identities $R\circ R=R$, $S\circ S=S$ give
$$
\begin{aligned}
T\circ T&=R\circ S\circ R\circ S\\
&\subseteq R\circ R\circ S\circ S
=R\circ S=T.
\end{aligned}
$$
For a direct chain description, start with $xRaSbRcSz$. Replace its middle $aSbRc$ by $aRdSc$ using condition (iii), then use <transitivity> inside $R$ and $S$ to obtain $xRdSz$. Hence $T$ is <transitive>. \b[Conditions (ii) and (iii) are equivalent.]
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