Equivalence of Definitions of Transitive Relation

Definition 1 implies Definition 2
Let $\RR$ be a relation which fulfills the condition:
 * $\tuple {x, y} \in \RR \land \tuple {y, z} \in \RR \implies \tuple {x, z} \in \RR$

Then:

Thus $\RR$ is transitive by definition 2.

Definition 2 implies Definition 1
Let $\RR$ be a relation that fulfills the condition:
 * $\RR \circ \RR \subseteq \RR$

$\RR$ does not fulfill the condition:
 * $\tuple {x, y} \in \RR \land \tuple {y, z} \in \RR \implies \tuple {x, z} \in \RR$

Then:

This contradicts our statement that $\RR \circ \RR \subseteq \RR$.

Hence by Proof by Contradiction $\RR$ does fulfills the condition:
 * $\tuple {x, y} \in \RR \land \tuple {y, z} \in \RR \implies \tuple {x, z} \in \RR$

Thus $\RR$ is transitive by definition 1.