Subgroup containment implies coset containment

Verbal statement
For left cosets: 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.

For right cosets: If one subgroup of a group is contained in another, then every right coset of the subgroup is contained in a right coset of the other subgroup.

Statement with symbols
For left cosets: Suppose $$H \le K \le G$$ are subgroups. Then, every fact about::left coset of $$H$$ is contained in exactly one left coset of $$K$$.

For right cosets: Suppose $$H \le K \le G$$ are subgroups. Then, every fact about::right coset of $$H$$ is contained in exactly one right coset of $$K$$.

Related facts

 * Coset containment implies subgroup containment: If a left coset of one subgroup is contained in a left coset of another, then the subgroup containment also holds.
 * 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: $$H \le K \le G$$, a left coset $$gH$$ of $$H$$ in $$G$$.

To prove: $$gH$$ is contained in a left coset of $$K$$ in $$G$$.

Proof: Since $$H \le K$$, any element in $$gH$$ is also an element in $$gK$$. $$gK$$ is a left coset of $$K$$, so we see that $$gH$$ is contained in a left coset of $$K$$.

It's also clear that any left coset of $$K$$ that contains $$gH$$ must be of the form $$gK$$, since it contains $$g$$.