Ring Homomorphism Preserves Subrings/Proof 1

Proof
Since $S \ne \varnothing$, $\phi \left({S}\right) \ne \varnothing$.

From Group Homomorphism Preserves Subgroups, $\left({\phi \left({S}\right), +_2}\right)$ is a subgroup of $\left({R_2, +_2}\right)$.

From Homomorphism Preserves Subsemigroups, $\left({\phi \left({S}\right), \circ_2}\right)$ is a subsemigroup of $\left({R_2, \circ_2}\right)$.

Thus, as $\left({R_2, +_2}\right)$ is a group and $\left({R_2, \circ_2}\right)$ is a semigroup, the result follows.