Codimension of intersection

Maybe there is some way for avoiding the homomorphism theorems, but they're so handy and powerful that it is better trying to understand them.

Consider the linear map $$ f\colon E\to E/E_1\oplus E/F_1,\qquad f(v)=(v+E_1,v+F_1) $$ The kernel of this map is $E_1\cap F_1$, so $f$ induces an injective linear map $$ \tilde{f}\colon E/(E_1\cap F_1)\to E/E_1\oplus E/F_1 $$ In particular, the domain is finite dimensional and $$ \dim E/(E_1\cap F_1)\le\dim(E/E_1\oplus E/F_1)= \dim(E/E_1)+\dim(E/F_1) $$ which is the same as saying that $$\DeclareMathOperator{\codim}{codim} \codim(E_1\cap F_1)\le\codim E_1+\codim F_1 $$

Hint:

Consider the diagram \begin{matrix} 0&\!\!\!\longrightarrow \!\!\!&E_1\cap F_1&\!\!\!\longrightarrow \!\!\!&E_1\bigoplus F_1&\!\!\!\longrightarrow \!\!\!&E_1+F_1&\!\!\!\longrightarrow &\!\!\!0\\ & & && \phantom{i\oplus j}\downarrow i\oplus j&& \phantom{k}\downarrow k\\ 0&\!\!\!\longrightarrow \!\!\!&\Delta E\!\!\!&\!\!\!\longrightarrow \!\!\!&E\bigoplus E&\!\!\!\longrightarrow \!\!\!&E&\!\!\!\longrightarrow &\!\!\!0 \end{matrix} where $\Delta E$ is the diagonal of $E\bigoplus E$, $i,j,k$ are the canonical injections, and the rightmost horizontal maps (from the direct sums) are $\; (x, y)\longmapsto x-y $.

You deduce first from this diagram there exists a map $f\colon E_1\cap F_1\longrightarrow \Delta E$ which makes the whole diagram commutative.

Apply the Snake's lemma to show $\operatorname{coker} f$ is (isomorphic to) a submodule of $E/E_1\bigoplus E/F_1$.