Exercise 13 from Roman's book "Advanced Linear Algebra" (page 107).
The author gives us a vector space $V$ with $$V=S_{1}\oplus T_{1}=S_{2}\oplus T_{2}$$ and asks us to prove that if $S_{1}$ and $S_{2}$ have finite codimension in $V$, then so does $\dim V/ (S_{1}\cap S_{2})$ and $$\dim V/ (S_{1}\cap S_{2})\leq \dim T_{1}+\dim T_{2}.$$
I did a lot of work but I didn't get anywhere. Thanks for your help.