Prove the third isomorphism theorem

I'm trying to prove the third Isomorphism theorem as stated below

Theorem. Let $G$ be a group, $K$ and $N$ are normal subgroups of $G$ with $K⊆N$. Then $$(G⁄K)⁄(N⁄K)≅G⁄N.$$

I look up for some answered on google, but I don't understand any of those. I wonder if any one can show me this proof step by step. I don't want to get it done, I want to understand why this theorem is true.

Solution 1:

Hints: define

$$\phi: G/K\;\to\; G/N\;,\;\;\phi(gK):=gN$$

(1) Prove $\,\phi\,$ is well a defined, onto homomorphism

2) Find $\,\ker\phi\,$

(3) Apply the first isomorphism theorem

Solution 2:

Hint: Show that $\phi(gK)=gN$ is a (well-defined) group homomorphism, and that its kernel is $N/K$. Then apply an (earlier) isomorphism theorem.