Coset containment implies subgroup containment

Verbal statement
If a fact about::left coset of one subgroup is contained in a left coset of another subgroup, then the subgroup is also contained in the other.

Statement with symbols
Suppose $$xH \subseteq yK$$ are cosets of subgroups $$H,K \le G$$. Then $$H \le K$$.

Related facts

 * Subgroup containment implies coset containment: If one subgroup of a group is contained in another, then every left coset of the subgroup is contained in a left coset of the other subgroup.
 * Nonempty intersection of cosets is coset of intersection: If the intersection of a collection of left cosets of subgroups is nonempty, it is a coset of the intersection of the corresponding subgroups.

Proof
Given: $$xH \subseteq yK $$ for some $$x,y \in G$$.

To prove: $$H \le K$$

Proof: Since $$x=xe$$ is in $$xH$$, which is contained in $$yK$$, there exists $$k_1 \in K$$ such that $$x=yk_1$$. Thus $$yk_1H \subseteq yK$$, which implies that there exist $$h \in H$$ and $$k_2 \in K$$ such that $$yk_1h=yk_2$$. Hence $$h=k_2k_1^{-1}$$, which is in $$K$$. So, $$H\le K$$.