Subset of Toset is Toset

Theorem
Let $\left({S, \preceq}\right)$ be a totally ordered set.

Let $T \subseteq S$.

Then $\left({T, \preceq \restriction_T}\right)$ is also a totally ordered set.

In the above, $\preceq \restriction_T$ denotes the restriction of $\preceq$ to $T$.

Proof
As $\left({S, \preceq}\right)$ is a totally ordered set, the relation $\preceq$ is a total ordering, and is by definition:


 * reflexive
 * antisymmetric
 * transitive
 * connected

From Properties of Restriction of Relation, a restriction of a relation which has all those properties inherits them all.

Thus $\preceq \restriction_T$ is also: and so is also a total ordering.
 * reflexive
 * antisymmetric
 * transitive
 * connected

Hence the result, by definition of totally ordered set.