Simplify F4 proof (FIRST VERSION)
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To simplify the map, you can remove two edges (not touching each other) of an F4. Since the other two faces, corresponding to the removed edges, may actually be the same face (if it goes around one of the other two neighbors of the F4), take care not to select such edges and select for removing the other two (this is provable as a consequence of the Jordan curve theorem). If you can color the map from which you removed the edges of the F4, when you restore the removed edges, will be always possible to find a color for the F4, since it will be surrounded by at most three colors (four neighbors two of which have the same color).