Let $G$ be a group, $H\leq G$, and $N\triangleleft G$ ($H$ is a subgroup, and $N$ is normal subgroup in $G$).
Let us define $H\cap N=K$
Let $\phi:H/K\rightarrow G/N$ be a map between these two quotient groups (I proved that $H\cap N$ is a normal subgroup in H, so the quotient group $H/K$ is indeed a group). $\phi$ defined as:
for every $aK\in H/K$; we have $\phi(aK)=aN$.
I also proved that $\phi$ is well defined, meaning that for every $a,b\in H$, if $aK=bK$, then $\phi(aK)=\phi(bK)$.
Now, there's only one last thing I need to prove: $\phi$ is injective, meaning that for every $aK, bK\in H/K$, if $\phi(aK)=\phi(bK)$, then $aK=bK$.
So far, this is my work;
If $\phi(aK)=\phi(bK)$ then it holds that $aN=bN$, so $a$ and $b$ are in the same coset of $N$, and so $an_0=bn_1$ for some $n_0, n_1\in N$.
And that is it, basically, (not much, I know) but I just couldn't think of anything that will lead me to $aK=bK$.
Any help would be greatly appreciated.