What I need to prove is a consequence of the following theorem.
Theorem A. Let $G$ be a finite $p$-group and suppose that its derived subgroup $G'$ is generated by 2 elements. Then there exists $x\in G$ s.t. $$ K_x(G):=\{[x,g]\mid g\in G \}=G'. $$
Now let $G$ be a pro-$p$ group whose derived subgroup is topologically generated by 2 elements. Define for any open normal subgroup $N$ of $G$ the set $$X_N:=\{x\in G\mid K_{xN}(G/N)=(G/N)'\}. $$ Notice that since $G$ is pro-$p$, every open normal subgroup of $G$ has finite index, therefore we can apply Theorem A to $G/N$ and conclude that $X_N\ne \emptyset$ for all $N\trianglelefteq_o G$.
What I need to prove is that the family $\{X_N\}_{N\trianglelefteq_o G}$ has the finite intersection property.
My guess is that we can take two any open normal subgroups $N$, $M$ of $G$, consider their product $NM$ which is still normal and open (since their product is equal to the union of cosets of an open subgroup) and prove that $$X_N\cap X_M \supseteq X_{NM}.$$ Since $NM\trianglelefteq_o G$, $X_{NM}$ would be non-empty and the statement would be proved.
All of this comes from here (at page 2 you can find Theorem A, as well as Theorem B whose proof, at page 11, is what I'm interested in). Notice that the author says that it's clear that $\{X_N\}_{N\trianglelefteq_o G}$ has the finite intersection property, so there could be an easier way to prove that.
from Hot Weekly Questions - Mathematics Stack Exchange
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