Variety of groups is congruence-permutable

Statement
The variety of groups is a congruence-permutable variety. In other words, any two congruences on any group permute, or every group is a congruence-permutable algebra in the variety of groups.

Translation to the language of groups
The translation of this statement to the ordinary language of groups is: given two normal subgroups of a group, their product is a normal subgroup. In other words, any two normal subgroups permute and their product is also a normal subgroup.

Proof in terms of other results
The result can be thought of as a combination of two facts:


 * Normal implies permutable
 * Normality is join-closed