Equivalence of Definitions of Total Ordering

Definition 1 implies Definition 2
Let $\RR$ be an ordering which is also connected.

By definition 2 of ordering:
 * $\RR \circ \RR = \RR$
 * $\RR \cap \RR^{-1} = \Delta_S$

and hence by definition of set equality:
 * $(1): \quad \RR \circ \RR \subseteq \RR$
 * $(2): \quad \RR \cap \RR^{-1} \subseteq \Delta_S$

By Relation is Connected and Reflexive iff Total, $\RR$ is a total relation.

Thus by Relation is Total iff Union with Inverse is Trivial Relation:
 * $(3): \quad \RR \cup \RR^{-1} = S \times S$

Hence $\RR$ is a total ordering by definition 2.

Definition 2 implies Definition 1
Let $\RR$ be a relation which fulfils the conditions:
 * $(1): \quad \RR \circ \RR \subseteq \RR$
 * $(2): \quad \RR \cap \RR^{-1} \subseteq \Delta_S$
 * $(3): \quad \RR \cup \RR^{-1} = S \times S$

By definition of transitive relation, we have from $(1)$ that $\RR$ is transitive.

From Relation is Antisymmetric iff Intersection with Inverse is Coreflexive, it follows from $(2)$ that $\RR$ is an antisymmetric relation.

By Relation is Total iff Union with Inverse is Trivial Relation it follows from $(3)$ that $\RR$ is a total relation.

From Relation is Connected and Reflexive iff Total, $\RR$ is both connected and reflexive.

Hence we have that $\RR$ is:
 * an ordering by definition 1

and hence:
 * a total ordering by definition 1.