Totally Ordered Set is Lattice

Theorem
Every totally ordered set is a lattice.

Proof
Let $$\left({S; \le}\right)$$ be a totally ordered set.


 * $$\forall x, y \in S: x \le y \lor y \le x$$
 * $$\forall x, y \in S: x \le y \Longrightarrow \sup \left\{{x, y}\right\} = y \land \inf \left\{{x, y}\right\} = x$$
 * $$\forall x, y \in S: y \le x \Longrightarrow \sup \left\{{x, y}\right\} = x \land \inf \left\{{x, y}\right\} = y$$

Thus the conditions for $$\left({S; \le}\right)$$ to be a lattice are fulfilled.