Union of Subgroups/Corollary 2

Theorem
Let $\struct {G, \circ}$ be a group.

Let $H, K \le G$. Let $H \vee K$ be the join of $H$ and $K$.

Then $H \vee K = H \cup K$ $H \subseteq K$ or $K \subseteq H$.

Proof
From the definition of join, $H \vee K$ is the smallest subgroup of $G$ containing $H \cup K$.

The result follows from Union of Subgroups.