Understanding a proof of a lemma to Jordan-Hölder Theorem. I have difficulty understanding the following lemma.

First, how do we know that $M\cap N={1}$, after replacing $M$ and $N$ as in the proof? 
Next, in the second part of the proof where it says we have deduced that $G \cong M \times N$, what it actually means is $G/(M\cap N) \cong M/(M\cap N) \times N/(M\cap N)$. By the Second Isomorphism Theorem, we can get the RHS to equal $MN/N \times MN/M$. But how do we get that this also equals $G/M \times G/N$ as in the conclusion?What we got from the proof in $MN=G$ is actually $MN/(M\cap N)=G/MN$. Can we conclude that $G=MN$ from this? I think this is the reason why the conclusion of the lemma follows but why is this true?
I would greatly appreciate any help in understanding this proof.
 A: For your first question: the proof explains it in its second paragraph - the triple $(G,M,N)$ is replaced by $(G/K,M/K,N/K)$, where $K=M \cap N$. So everything from then on is done modding out by $K$. Write overbar $\bar{.}$ for modding out by $K$. In $\bar{G}$ we have $\bar{M} \cap \bar{N}=\bar{1}$. So this is why one can assume that $M$ and $N$ intersect trivially. Now you can proceed with the overbar, but it saves writing if that is omitted and just assume $M$ and $N$ intersect trivially and proceed with the proof.  But, and this is maybe where your misunderstanding comes from, $G=MN$ irrespective of $M \cap N$ being trivial or not. The proof of this ("the claim") is slightly misplaced. I agree, this could had been proved right at the start and not somewhere in the middle where it is assumed that $M$ and $N$ intersect trivially. That has nothing to do with it!
Now for your second remark one uses $G=M \times N$, of which the proof says "That specific statement is only right with the additional tool of $M \cap N=1$". So you only get $\bar{G}=\bar{M} \times \bar{N}$, whence $G/(M \cap N) \cong M/(M \cap N) \times N/(M \cap N)$, and since $G=MN$, $ M/(M \cap N) \cong G/N$ and symmetrically $N/(M \cap N) \cong G/M$. 
I hope you are convinced now! 
