Here is a (wrong) statement: * [the sum S of two even numbers A1 and A2] is an odd number Here is its negation: * [the sum S of two even numbers A1 and A2] i

Here is a (wrong) statement: * [the sum S of two even numbers A1 and A2] is an odd number Here is its negation: * [the sum S of two even numbers A1 and A2] is not an odd number or alternatively: * [the sum S of two even numbers A1 and A2] is an even number. Let's "you can't prove the negative": * A1 is even => there exists an integer a1 such that A1 = 2a1 A2 is even => there exists an integer a1 such that A2 = 2a2 by substitution the sum (A1+A2) = (2a1 + 2a2) * by distributivity: (A1+A2)=2(a1+a2) sum s of integers a1,a2 is an integer: a1 + a2 = s integer * substitution (A1+A2)=2s with s integer S=(A1+A2)=2s, hence S is even an even number is not odd * hence [the sum S of two even numbers A1 and A2] is not odd We proved a theorem that was also a negation of a statement!!