in proving that P(x,y) holds for every x and y, one may assume "without loss of generality" that x <= y. There is no loss of generality in this assumption, since once the case x <= y => P(x,y) has been proved, the other case follows by interchanging x and y: y <= x => P(y,x), and by symmetry of P, this implies P(x,y), thereby showing that P(x,y) holds for all cases.

On the other hand, if neither such a symmetry nor another form of equivalence can be established, then