Four color theorem and map pre-simplification of faces with less than 5 edges
The result is known since Kempe in 1879, and is mentioned in http://en.wikipedia.org/wiki/Four_color_theorem as "Kempe also showed correctly that G can have no vertex of degree 4...."  Your proof does not work, because when you remove two edges joining face B with A and C, A and C may already have been adjacent.  Since you give the combined face a color, then return face B with a new color, you leave A and C adjacent and of the same color.