**"Lemma 4": If we can color every
regular map on the sphere with five or fewer colors then we can
color any map on the sphere with five or fewer colors.**
####

**Proof:** Consider any map on the sphere and a vertex,
A, that is not regular.

1. *Place a small circular region about the vertex*
making a new graph with new edges and vertices determined by the intersection
of the circle with the original graph.

2. Do step 1 for every vertex on the original map that
is not regular. The *resulting map is regular*.

3. By the assumption, *the
regular map can be colored* with five or fewer colors, so color it.

4.Now *remove the circles*, coloring the sectors
of the original map determined by the added circles by the color of the
adjacent region from the regular map.

*This colors the original map* with the no
more than the same number of colors as the regular map.