The Love Reciprocated Demo
Suppose that everyone loves someone (that is, '(^x)(%y)Lxy' is true on one obvious interpretation of 'L'). And that if one person loves a second, that love is reciprocated ('(^x)(^y)(Lxy>Lyx)'). Too ideal, of course! But given these two premises, it follows that everyone is loved by someone. ('(^x)(%y)Lyx'; note that 'y' and 'x' have been swapped from their positions in the first premise.)
First, see if you can make sense of the above, then do the derivation...or perhaps you should do the derivation first, then see if you can comprehend the first paragraph above!