Way Below is Congruent for Join

Theorem
Let $L = \left({S, \vee, \preceq}\right)$ be a join semilattice.

Then $\ll$ is congruence relation for $\vee$:
 * $\forall a, b, x, y \in S: a \ll x \land b \ll y \implies a \vee b \ll x \vee y$

where $\ll$ denotes the way below relation.

Proof
Let $a, b, x, y \in S$ such that $a \ll x$ and $b \ll y$

By Join Succeeds Operands:
 * $x \preceq x \vee y$ and $y \preceq x \vee y$

By Preceding and Way Below implies Way Below and definition of reflexivity:
 * $a \ll x \vee y$ and $b \ll x \vee y$

Thus by Join is Way Below if Operands are Way Below:
 * $a \vee b \ll x \vee y$