Left and Right Coset Spaces are Equivalent/Proof 1

Proof
Let there be exactly $r$ different left cosets of $H$ in $G$.

Let a complete repetition-free list of these left cosets be:
 * $a_1 H, a_2 H, a_3 H \ldots, a_r H: a_1, a_2, \ldots, a_r \in G$

From Coset Spaces form Partition, every element of $G$ is contained in exactly one of the left cosets.

Let $x \in G$.

Then, for $1 \le i \le r$:

Since $x^{-1} \in a_i H$ is true for precisely one value of $i$, it follows that $x \in H a_i^{-1}$ is also true for precisely that value of $i$.

So there are exactly $r$ different right cosets of $H$ in $G$, and a complete repetition-free list of these is:


 * $H a_1^{-1}, H a_2^{-1}, H a_3^{-1}, \ldots, H a_r^{-1}$

The result follows.