IHOP: Assume it is true for \(h\) vertices. Let \(K\) be a planar graph with \(h+1\) vertices. By our lemma** we know that there will be a vertex with a degree at most 5. Now suppose that we remove this vertex and have \(K-v\)(vertex). Then we still have a planar graph. With the inductive hypothesis then, the statement is colorable with 6 colors. If we bring back \(v\) to the graph, since it has degree at most 5 and we have 6 colors, we can assign \(v\) a color to properly color the graph. Hence, the statement is true.