Minimum size of generating set of quotient is less than or equal to that of whole group

Statement
Suppose $$G$$ is a group, $$H$$ is a normal subgroup, and $$K = G/H$$ is the fact about::quotient group. Then, the fact about::minimum size of generating set for $$K$$ is less than or equal to the minimum size of generating set for $$G$$, where the comparison is done as (possibly infinite) cardinals.

In particular, if $$G$$ is a finitely generated group, so is $$K$$.

Related facts

 * Minimum size of generating set of subgroup may be more than of whole group
 * Minimum size of generating set of direct product of two groups is bounded by sum of minimum size of generating set of each factor