Equivalence of Definitions of Ordering/Proof 1

Theorem
The following definitions of ordering are equivalent:

Definition 1 implies Definition 2
Let $\mathcal R$ be a relation on $S$ satisfying:

By Reflexive and Transitive Relation equals Composite with Itself:
 * $\mathcal R \circ \mathcal R = \mathcal R$

By Relation is Antisymmetric and Reflexive iff Intersection with Inverse equals Diagonal Relation:
 * $\mathcal R \cap \mathcal R^{-1} = \Delta_S$

Thus $\mathcal R$ is an ordering by definition 2.

Definition 2 implies Definition 1
Let $\mathcal R$ be a relation which fulfils the conditions:
 * $(1): \quad \mathcal R \circ \mathcal R = \mathcal R$
 * $(2): \quad \mathcal R \cap \mathcal R^{-1} = \Delta_S$

By definition of set equality, it follows from $(1)$:
 * $\mathcal R \circ \mathcal R \subseteq \mathcal R$

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

By Relation is Antisymmetric and Reflexive iff Intersection with Inverse equals Diagonal Relation, it follows from $(2)$ that:
 * $\mathcal R$ is reflexive
 * $\mathcal R$ is antisymmetric.

Thus $\mathcal R$ is an ordering by definition 1.