Equivalence of Definitions of Reflexive Relation

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

Suppose $\Delta_S \not \subseteq \mathcal R$.

Then by definition of Diagonal Relation
 * $\exists \left({x, x}\right) \in S \times S: \left({x, x}\right) \notin \mathcal R$

Thus:
 * $\exists x \in S: \left({x, x}\right) \notin \mathcal R$

From this contradiction it follows that:
 * $\Delta_S \subseteq \mathcal R$

Thus $\mathcal R$ is reflexive by Definition 2.

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

Then:

Thus $\mathcal R$ is reflexive by Definition 1.