Let be subgroups.

Define to be the cosets of .

Define to be the cosets of .

Define and .

Note that is a subgroup.

Define a mapping from to by .

We need to show this mapping is well-defined.

To show this we use a fact that if then for some subgroup.

The next thing we notice is that this mapping is one-to-one.

For if .

Since this mapping is one-to-one it means .