Let S be a subgroup of the group G. Suppose that a, b belongs to G satisfies Sa = bS. Thus the right coset of S containing a equals the left coset of S containing b.
Show that Sa = aS = bS = Sb.
from $\displaystyle Sa=bS,$ we get $\displaystyle a=bs,$ and $\displaystyle b=ta$ for some $\displaystyle s, t \in S.$ thus: $\displaystyle aS=bsS=bS=Sa=Sta=Sb.$