Equivalence of Definitions of Transitive Relation

Theorem
The following definitions of transitive relation are equivalent:

Definition 1 implies Definition 2
Let $\mathcal R$ be a relation which fulfils the condition:
 * $\left({x, y}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R \implies \left({x, z}\right) \in \mathcal R$

Then:

Thus $\mathcal R$ is transitive by definition 2.

Definition 2 implies Definition 1
Let $\mathcal R$ be a relation which fulfils the condition:
 * $\mathcal R \circ \mathcal R \subseteq \mathcal R$

Suppose $\mathcal R$ does not fulfil the condition:
 * $\left({x, y}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R \implies \left({x, z}\right) \in \mathcal R$

Then:

From this contradiction it follows that $\mathcal R$ does fulfil the condition:
 * $\left({x, y}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R \implies \left({x, z}\right) \in \mathcal R$

Thus $\mathcal R$ is transitive by definition 1.