Sum of Ideals is Ideal/Corollary

Corollary to Sum of Ideals is an Ideal
Let $J_1$ and $J_2$ be ideals of a ring $\left({R, +, \circ}\right)$.

Let $J = J_1 + J_2$ be an ideal of $R$ where $J_1 + J_2$ is as defined in subset product.

Then:
 * $J_1 \subseteq J_1 + J_2$
 * $J_2 \subseteq J_1 + J_2$

Proof
From Sum of Ideals is an Ideal we have that $j$ is an ideal of $R$.

Then:
 * $0_R \in J_1 + J_2$

and so:
 * $\forall x \in J_1: x + 0_R = x \in J_1 + J_2$

Similarly for $J_2$.