Join Semilattice is Ordered Structure/Proof 2

Proof
Let $a, b, c \in S$.

Let $a \preceq b$.

By the definition of join semilattice:
 * $a \vee b = b$

Thus:
 * $\paren {a \vee b} \vee c = b \vee c$

Since $\vee$ is associative, commutative, and idempotent:
 * $\paren {a \vee c} \vee \paren {b \vee c} = b \vee c$

Therefore, $a \vee c \preceq b \vee c$.

From Join is Commutative, we conclude that:
 * $c \vee a \preceq c \vee b$