Equivalence of Definitions of Ordering/Proof 1

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

By Reflexive and Transitive Relation is Idempotent:
 * $\RR \circ \RR = \RR$

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

Thus $\RR$ is an ordering by definition 2.

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

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

Thus, by definition, $\RR$ is transitive.

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

Thus $\RR$ is an ordering by definition 1.