= Solution
A <binary relation> $T$ is transitive exactly when $T\circ T\subseteq T$. If condition (iii) holds, associativity and monotonicity of <composition of relations> give
$$
T\circ T=R\circ S\circ R\circ S
\subseteq R\circ R\circ S\circ S
=R\circ S=T.
$$
Hence (iii) implies (ii). Conversely, the inclusions $R,S\subseteq T$ imply
$$
S\circ R\subseteq T\circ T.
$$
If $T$ is a <transitive relation>, the right side lies in $T$, which is condition (iii). Thus \b[(ii) and (iii) are equivalent].
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