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edges, it follows that G’ has n edges, and by the inductive hypothesis |V’|-|E’|+|F’| = 2. Based on how G’ was constructed, we have the relationships that |V’|=|V|, |E’|=|E|-1, and |F’|=|F|-1. Substituting these values into the equation, |V|-(|E|-1)+|F|-1 = |V|-|E|+|F|=2. Since Euler’s Theorem is