Union of Subgroups/Corollary 1

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

Let $H, K \le G$. Let $H \cup K$ be a subgroup of $G$.

Then either $H \subseteq K$ or $K \subseteq H$.

Proof
neither $H \subseteq K$ nor $K \subseteq H$.

Then from Union of Subgroups it follows that $H \cup K$ is not a subgroup of $G$.

The result follows by Proof by Contradiction.