I take to be . That seems to be how your other posts define the naturals.
Proof: .
The ( ) direction is obvious, just multiply both sides of the left hand side by and you get the right.
For the converse, suppose . Since we know that , thus we can divide by it. Dividing throughout the inequality by yeilds the desired result.
QED
You may see my response to your other posts for a more rigorous write-up. The language I used here is not very appropriate. This is very similar to the proof I did.
x > y if and only if x = y + k has a solution in N for k (by definition). Then zx > zy if and only if zx = zy + j has a solution for some j in N. If you multiply x = y + k by z you get zx = zy + kz, thus choosing j = kz we have the solution to the equation.