Understanding the proof of a criterion of equivalent group extensions

Question

I stuck in a step of the proof of the equivalence of two group extensions for 30 mins. In the place of red arrow, how does the previous line deduce the next?

PS: The author denote the operation in $G,~G'$ as $+$ for convenience (even outside the abelian group $K$).

Answer 1

The fact that you have a group extension means that if $g\in G'$ and $k\in K$ then inside $G'$ you have $$g+k=p'(g)k + g.$$

So if $g=l'(x)$, you get $$l'(x)+k=xk + l'(x).$$

Apply this to $k=b+h(y)$.