Apostol's Field Axioms
i'm not sure which "axiom" you're talking about, is it:
1/(xy) = (1/x)(1/y)?
one doesn't need "uniqueness" of an inverse to show that, just "existence":
(xy)[(1/x)(1/y)] = x(y[(1/x)(1/y)] (associativity)
= x(y[(1/y)(1/x)] (commutativity)
= x([y(1/y)](1/x)] (associativity)
= x((1)(1/x)) (using y(1/y) = 1, since 1/y is "a" inverse for y)
= x(1/x) (multiplicative identity property of 1)
= 1 (since 1/x is "a" inverse for x).
thus 1/(xy) is "one" inverse for xy (it might be, that there are others, but at the very least we know for sure that (xy)(1/(xy)) = 1 if x(1/x) = 1, and y(1/y) = 1).
what is going on is this:
the actual process of computing a/b + c/d goes like this:
1. a/b + c/d = (a/b)(1) + (c/d)(1) =
2. (a/b)(d/d) + (c/d)(b/b) =
3. (ad/(bd)) + (bc/(bd)) =
4. (ad + bc)/(bd)
1-->2 is obvious
2-->3 uses commutativity and associativity of multiplication
(which is why writing 1/x = x-1 makes it clearer)
3-->4 uses the distributive law
for the above to work, we just need the existence of a-1 and b-1, we don't need uniqueness (the possibility of division).
of course, uniqueness is easy to prove given existence:
suppose ab = ba = 1, and ca = ac = 1:
then c = c(1) = c(ab) = (ca)b = (1)b = b.