Let be any subgroup of the group . The set of left cosets of in form a partition of . Furthermore, if and only if and in particular, if and only if and are representatives of the same coset.
Therefore so .
To show that distinct left cosets are disjoint, suppose and show that .
Let Then for some .
Multiply both sides of the second equation by
Now I can write as , so for some ,
This means .
Similarly and so . This finishes the first part of the proof.
To show if and only if ,
which means for some .
Since , if and only if and are representatives of the same coset.