Hi guys,

Suppose following axiomatization of real numbers:

( is syntactical sugar for uniqueness quantifier)

(+ 1)

(+ 2)

(+ 3)

(. 1)

(. 2)

(. 3)

(+ .)

(< 1) forall forall , one of the following holds: , ,

(< 2)

(+ <)

(. <)

(sup) Let be non-empty set, s.t. it has upper bound. Then

Now following theorem holds: There is precisely one element (which will be denoted ), which is solution of equation .

Proof: Assume some fixed . Let the only solution of be denoted by symbol . So holds.

Now proof is straightforward and I will not finish it. My problem is that is formal symbol of our first-order theory (i.e. it is non-logical constant) and no axiom define any property that should obbey. However the first step of the theorem says something like . My question is why is this correct? I don't see any inference rule which interlinks and axiom (+ 3).

any ideas why is that correct? thanks