isomorphism between two random vector subspaces

Question

I have this problem which is the following : Let K be a field, $E$ a K-vector space (finite or infinite dimension) and $F$,$G$ two vector subspaces of E. Now, consider S to be a common supplementary subspace of $F, G$ in $E$, that is : $E = F \bigoplus S = G \bigoplus S$.
I wish to show that $F \cong G$. Here is my attempt of solution: note that in the case of $E$ being of finite dimension, via Grassman formula we get equality of dimensions and then it's finished.
Let $f: E \rightarrow E$ be a mapping such that $f(x_1 + s) = x_2 +s$ where $(x_1,x_2,s) \in F \times G \times S$. We can see that $f$ is a bijective linear mapping. Let us consider $u$ the restriction of $f$ to $F$. Then $u : F \rightarrow Im(u)$. Though, we can now show that $Im(u) \subset G$. Let $x_1 \in F$. It follows that $u(x_1)=u(x_1+0)=x_2 \in G$. But then since $f$ is a linear bijective mapping, it is also the case for $u$. We have proved that $f: F \rightarrow G$ is an isomorphism of vector spaces, so $dim F = dim G$, giving $F \cong G$.
I cannot find the same exercise on the internet so I'm not really able to check if it's the correct way to go. Please let me know if it is correct or not and if not where are the flaws! Thank you