At least as many disks as regions

Question

I am looking for reference of proof for the following fact:

Given a set $D$ of same radius disks, embedded in the plane, it holds that the number of connected regions in $\mathbb{R}^2 \setminus \cup_{d \in D}$ is at most $|D|$, i.e., there can not be more connected regions than disks.

Does anybody know of such a proof?

Answer 1

The right half of this image:

shows 42 disks, whose removal leaves 61 regions. It would suffice to take this many disks:

O O O O O 
 O O O O O
O O O O O

in which case there would be 17 regions, and 15 disks.