Linear isomorphism of quotient spaces

Question

Need some help.

Let $M,N$ subspaces of L. Prove that the following map is a linear isomorphism.

$$\phi:\frac{M+N}{N}\longrightarrow\frac{M}{M\cap N}$$ Defined by $$\phi(m+n+N)=m+M\cap N$$

I'm stucked

Answer 1

If $m+M\cap N=m'+M\cap N$, then $m-m'\in M\cap N$ so $m-m'\in N$ ($m'+N=m+N$) and $\phi$ is injective. The map is also surjective since $\phi(m+N)=m+M\cap N$.