For example, consider a rather trivial theorem which says that if you have two positive integers in decimal representations Aaa...aa and Bbb..bb, of same length

For example, consider a rather trivial theorem which says that if you have two positive integers in decimal representations Aaa...aa and Bbb..bb, of same length, then the sum of them is larger than number (A+B)00..00. So this theorem gives you a property of the result (the result is larger than some number) without you having to calculate the result (sum the numbers) explicitly. The Hashlife algorithm mentioned in this thread is another case, basically it's an algorithm to prove certain "theorems" about the Game of Life and applying those theorems to speed it up.