Index is multiplicative for profinite groups

Statement
Suppose $$G$$ is a profinite group and $$K,H$$ are closed subgroups of $$G$$ with $$K \le H$$. Note that $$K$$ automatically becomes closed in $$H$$. (Conversely, $$K$$ being closed in $$H$$ and $$H$$ being closed in $$G$$ would imply that $$K$$ is closed in $$G$$). Then, we have:

$$[G:K] = [G:H][H:K]$$

where $$[G:K]$$ denote the respective values for the index of a closed subgroup in a profinite group where the subgroup is $$K$$ and the group is $$G$$. Similarly for $$[G:H]$$ (index of subgroup $$H$$ in group $$G$$) and $$[H:K]$$ (index of subgroup $$K$$ in group $$H$$).

Related facts

 * Index is multiplicative
 * Lagrange's theorem for profinite groups
 * Lagrange's theorem