I have been trying to work through a proof for Lagrange's theorem extended to profinite groups from John Wilson's book.
The statement of the theorem is "for $H$ and $K$ subgroups of a profinite group $G$, such that $K\leq H \leq G$ then $\lvert G:K\rvert=\lvert G:H\rvert \cdot \lvert H:K\rvert$.
The proof begins by stating that if $N\triangleleft_O G$ is an open normal subgroup then $$\lvert G:NK\rvert=\lvert G:NH\rvert \cdot \lvert NH:NK\rvert=\lvert G:NH\rvert \cdot \lvert H:(N\cap H)K\rvert$$ and $N\cap H\triangleleft_O G$.
I follow the proof up until here but the bit which gets me is that it then says that therefore $\lvert G:K\rvert$ divides $\lvert G:H\rvert \cdot \lvert H:K\rvert$. I believe this has something to do with a previous definition that the index of a subgroup $H\leq G$ for a profinite group $G$