Lagrange's theorem for profinite groups

Statement
Suppose $$G$$ is a profinite group and $$H$$ is a closed subgroup of $$G$$. Then, we have:

$$|G| = |H|[G:H]$$

where:


 * $$|G|$$ denotes the order of $$G$$ in the sense of order of a profinite group, and is a supernatural number.
 * $$|H|$$ denotes the order of $$H$$ in the sense of order of a profinite group, and is a supernatural number.
 * $$[G:H]$$ denotes the index of $$H$$ in $$G$$ in the sense of index of a closed subgroup in a profinite group, and is a supernatural number
 * The multiplication on the right is in the sense of multiplication of supernatural numbers.

Related facts

 * Index is multiplicative for profinite groups
 * Lagrange's theorem is the ordinary version.