Equivalence of Definitions of Total Ordering

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

By definition 2 of ordering:
 * $(1): \quad \mathcal R \circ \mathcal R = \mathcal R$
 * $(2): \quad \mathcal R \cap \mathcal R^{-1} = \Delta_S$

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

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

Hence $\mathcal R$ is a total ordering by definition 2.

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

By Relation is Total iff Union with Inverse is Trivial Relation it follows that $\mathcal R$ is a total relation.

From $(1)$ and $(2)$, $\mathcal R$ is an ordering by definition 2.

Thus by definition 1 of ordering, $\mathcal R$ is a reflexive relation.

From Relation is Connected and Reflexive iff Total, $\mathcal R$ is a connected relation.

Hence $\mathcal R$ is a total ordering by definition 1.