This paper presents detailed analyses of congruences modulo in the case of even sum S = A + B. These analyses were performed in order to design a way to demonstrate GSC in an elementary logical way. Even if we succeed with the rules of congruence in putting an even number in the sum of two prime numbers, this does not constitute a definitive mathematical proof, which is why the GSC remains unprovable. This is why we must resort to a logical reasoning which consists of eliminating false propositions and keeping only one which is true. The one which is true must lead us to the truth of the GSC and thus we succeed in demonstrating it mathematically. This paper provides an elementary mathematical proof by deciding between four propositions such that the GSC is the only true one (logical reasoning by an indirect proof). This conclusion is reached by taking into account established facts in mathematics about prime numbers in [0 - n] and [n - 2n] intervals.