Right Cosets are Equal iff Left Cosets by Inverse are Equal

Theorem
Let $G$ be a group whose identity is $e$.

Let $H$ be a subgroup of $G$.

Let $g_1, g_2 \in G$.

Then:
 * $H g_1 = H g_2 \iff {g_1}^{-1} H = {g_2}^{-1} H$

where:
 * ${g_1}^{-1}$ and ${g_2}^{-1}$ denote the inverses of $g_1$ and $g_2$ in $G$
 * $H g_1$ and $H g_2$ denote the right cosets of $H$ by $g_1$ and $g_2$ respectively
 * ${g_1}^{-1} H$ and ${g_2}^{-1} H$ denote the left cosets of $H$ by ${g_1}^{-1}$ and ${g_2}^{-1}$ respectively.