In this manuscript we present a mathematical proof in the traditional manner for the Four-Color Problem, without using any software program. We do so in two parts. In Part One we consider the four-color problem with the following constraint. Each country is a connected region. A proof using mathematical induction on the number of countries of a map is given. In Part Two we prove the Four-Color Problem without this constraint. No attempt is made to generalize the original four-color problem. MSC 2010 Code: 05-02.