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!!

Came to the comments to make sure someone had already mentioned this!