I encountered the following implication in a text book:

which I'm strugling to understand...

Being (a) = f(x,y), it can be derived integrating the first equation:

That's because f(x,x) := 0 by definition.

Therefore the initial implication does not make sense to me.

Does it sense to some you guys?

If yes, would you pls help me to understand/derive it?

Thanks!