Subtraction on Numbers is Anticommutative/Integral Domains

Theorem
The operation of subtraction on the numbers is anticommutative.

That is:
 * $a - b = b - a \iff a = b$

Proof
Let $a, b$ be elements of one of the standard number sets: $\Z, \Q, \R, \C$.

Each of those systems is an integral domain, and so is closed under the operation of subtraction.

Necessary Condition
Let $a = b$.

Then $a - b = 0 = b - a$.

Sufficient Condition
Let $a - b = b - a$.

Then:

We have that:
 * $a - b = b - a$

So from the above:
 * $b - a = - \paren {b - a}$

That is:
 * $b - a = 0$

and so:
 * $a = b$

Also see

 * Definition:Natural Number Subtraction