Principal Ideal Domain is Bézout Domain

Theorem
Let $D$ be a principal ideal domain.

Then $D$ is a Bézout domain.

Proof
Let $J_1, J_2$ be ideals of $D$.

From Sum of Ideals is Ideal, $J_1 + J_2$ is an ideal of $D$.

By definition of principal ideal domain, $J_1 + J_2$ is principal.

Hence $D$ is a Bézout domain.