The statement that any map can be colored using only four colors in such a way that no two adjacent regions have the same color is known as the Four Color Theorem. This theorem was first stated in 1852 by Francis Guthrie and later proven by Kenneth Appel and Wolfgang Haken in 1976, with the help of a computer program.

The proof of the Four Color Theorem is complex and lengthy, but here is a basic outline of the approach used by Appel and Haken:

1. Convert the given map into a planar graph, where each region of the map corresponds to a vertex, and adjacent regions are connected by an edge.

2. Reduce the planar graph to a simpler form, called a reduced graph, by applying a series of transformations known as discharging rules. These rules involve redistributing the colors assigned to the vertices based on certain criteria.

3. Show that every vertex in the reduced graph satisfies certain necessary conditions, known as the Euler's formula, which relate the number of vertices, edges, and faces of a planar graph.

4. Utilize a computer program to perform an exhaustive analysis of all possible configurations of the reduced graph, also known as discharging sequences. This involves checking thousands of cases to determine if any of them violate the necessary conditions established in step 3.

5. Conclude that since no violations of the necessary conditions are found in any discharging sequence, the Four Color Theorem holds true for all possible maps.

It is important to note that the proof of the Four Color Theorem by Appel and Haken is not considered a traditional, easily verifiable mathematical proof due to the significant computational element involved. However, since its publication, numerous mathematicians and experts in the field have examined the proof and generally accepted its validity.

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