Given any r in positive R (reals), the number sqroot(r) is unique in the sense that, if x is a positive real number such that x^2 = r, then x = sqroot(r)
Any help with this proof would be appreciated.
First, the question was about reals, so ordered fields and stuff is way too much for this EXCEPT for the part that we talk about POSITIVE numbers.
Second, the equalities x^2 = y^2 <==> (x-y)(x+y) = 0 are true in any commutative ring, and if we add some other condition, not necessarily of ordered
fields, we can decide whether x = y or x = -y (for example, if we take the usual representatives of the residue classes
modulo a prime p, we can decide that the solution to x^2 = has to be between
0 and (p-1)/2 (mod p), say...