Minimum size of generating set of quotient group is at most minimum size of generating set of group

Statement
Suppose $$G$$ is a group, $$N$$ is a normal subgroup, and $$G/N$$ is the corresponding quotient group. Then, the minimum size of generating set of $$G/N$$ is at most equal to the minimum size of generating set of $$G$$. 

Related facts

 * Minimum size of generating set of subgroup may be strictly greater than minimum size of generating set of group
 * Minimum size of generating set of extension group is bounded by sum of minimum size of generating set of normal subgroup and quotient group
 * Minimum size of generating set of direct product is bounded by sum of minimum size of generating set of each factor
 * Minimum size of generating set of direct product is bounded below by maximum of minimum size of generating set of each factor