Totally Ordered Set is Lattice

Theorem
Every totally ordered set is a lattice.

Proof
Let $\struct {S, \preceq}$ be a totally ordered set.

Then we have:


 * $\forall x, y \in S: x \preceq y \lor y \preceq x$
 * $\forall x, y \in S: x \preceq y \implies \sup \set {x, y} = y \land \inf \set {x, y} = x$
 * $\forall x, y \in S: y \preceq x \implies \sup \set {x, y} = x \land \inf \set {x, y} = y$

Thus the conditions for $\struct {S, \preceq}$ to be a lattice are fulfilled.