In the Wikipedia article about the four colour theorem it is stated
First, if planar regions separated by the graph are not triangulated, i.e. do not have exactly three edges in their boundaries, we can add edges without introducing new vertices in order to make every region triangular, including the unbounded outer region. If this triangulated graph is colorable using four colors or fewer, so is the original graph since the same coloring is valid if edges are removed.
How is the same coloring valid if we remove edges? This will make a union of regions with different colours and neighboring several other regions. How can we choose the same coloring and if there is some process to do so, how will it be valid?