Cosets Partition a Group

Cosets Partition a Group: Let $G$ be a group and let $H\le G$ be a subgroup . Then the set of left cosets $\{gH : g\in G\}$ forms a partition of $G$ (and similarly for right cosets $\{Hg:g\in G\}$).

More precisely: every $g\in G$ lies in the left coset $gH$, and any two left cosets are either equal or disjoint.

Proof sketch: If $x\in gH\cap g'H$, then $x=gh=g'h'$ for some $h,h'\in H$, hence $g^{-1}g'=hh'^{-1}\in H$ and so $g'H=gH$. Thus distinct cosets cannot intersect.