let's see if we can prove it:

f(x-y) = f(x+(-y)) = f(x)+f(-y).

does f(-y) = -f(y)? well, let's see if we can prove it:

f(y) + f(-y) = f(y+(-y)) = f(0) = 0, so f(-y) must be the additive inverse of f(y), -f(y).

therefore (going back to our first proof, now) f(x-y) = f(x) + f(-y) = f(x) + -f(y) = f(x) - f(y). yep, it must be true.