Assume x is greater than y by z and y is greater than x by z (seems impossible, and it might very well be, but read on)
x> or equal to y
y> or equal to x
x=y+z
y=x+z
y=y+z+z
0=2z
0=z
x=y+z
x=y+0
x=y
I think what I just said is that if x is greater than y and y is greater than x, they must be greater than eachother by zero, thus proving they are equal. It also works if x is greater than y by z and y is greater than x by say q (though in the end both q and z are zero).
Does this make any sense?
Not unless you're using a strange definition of "greater than" (or you used the wrong symbol). Are you familiar with the Law of Trichotomy?
The fact that x = y + z for z ≥ 0 is equivalent to x ≥ y, not x > y. You proved correctly that if x ≥ y and y ≥ x, then x = y. This fact is expressed by saying that ≥ is antisymmetric. In contrast, x > y and y > x is impossible. This means that > is asymmetric.