Implication in Zassenhaus Lemma I need to proof Zassenhaus Lemma using the First Isomorphism Theorem and I have a problem with the following implication:

$H' \vartriangleleft H < G, K' \vartriangleleft K < G \Longrightarrow H'(H \cap K') \vartriangleleft H'(H \cap K) < H.$

I have shown that $H \cap K' \vartriangleleft H \cap K, $ but I don't see why $H'(H \cap K') \vartriangleleft H'(H \cap K)$. It should follow from the First Isomorphism Theorem and from $H' \vartriangleleft H,$ but I don't see how.

Perhaps it is more clear if you prove the following:

Claim: If $C \trianglelefteq H$ and $A \trianglelefteq B \leq H$, then $CA \trianglelefteq CB \leq H$.

Then your claim follows with $C = H'$ and $A = H \cap K'$, $B = H \cap K$. Note that in this situation $A \trianglelefteq B$  since $K' \trianglelefteq K$.
Maybe you want to try to prove the claim yourself. If you get stuck, here is a proof:

 $CA$ and $CB$ are subgroups since $C$ is normal, and $CA \leq CB$ is obvious. Now $B$ normalizes $CA$ since $C$ is normal, and since $A \trianglelefteq B$. Furthermore, $C$ normalizes $CA$ since $C$ is a subgroup of $CA$. Thus $CB$ normalizes $CA$, that is, $CA \trianglelefteq CB$.

