Partial coloring using a specific sequence of given 4 colors, followed by another specific (but different from the previous) sequence of given 4 colors (to be used for the next row of countries), known as a unit pattern, can follow simple rules to color the whole map (of the countries shaped as squares of equal size) guaranteeing that no matter how large the map gets (to infinity in all four directions), there will be no two countries sharing the same color, thus proving that the theorem is valid.
Given a=blue, b=green, c=red, d=yellow The unit pattern is
a b c d c d a b
This is like looking at a map from above and seeing 8 countries colored in colors denoted by a(=blue), b(=green), c(=red), d(=yellow). It's the same as saying,