Elements of Module with Equal Images under Linear Transformations form Submodule

Theorem
Let $G$ and $H$ be $R$-modules.

Let $\phi$ and $\psi$ be linear transformations from $G$ into $H$.

Then the set $S = \left\{{x \in G: \phi \left({x}\right) = \psi \left({x}\right)}\right\}$ is a submodule of $G$.

Also see
Compare with Homomorphisms on the Same Groups.