Cosets are Equivalent/Proof 2

Theorem
All left cosets of a group $G$ with respect to a subgroup $H$ are equivalent.

That is, any two left cosets are in one-to-one correspondence.

The same applies to right cosets.

As a special case of this:
 * $\forall x \in G: \left|{x H}\right| = \left|{H}\right| = \left|{H x}\right|$

where $H$ is a subgroup of $G$.

Proof
Follows directly from Set Equivalence of Regular Representations.