On residually transcendental valued function fields of conics

Let K/Ko be a finitely generated field extension of transcendence degree 1. Let u<SUP>0</SUP> be a valuation of K<SUB>0</SUB> and v a valuation of K extending v<SUB>0</SUB> such that the residue field of vis a transcendental extension of the residue field k<SUP>0</SUP> of v<SUP>0</SUP>/such a prolongation vwill be called a residually transcendental prolongation of v<SUP>0</SUP>. Byan element with the uniqueness propertyfor (K, v)/(K<SUP>0</SUP>, v<SUP>0</SUP>) (or more briefly for v/v<SUP>0</SUP>)we mean an element / of K having u-valuation 0 which satisfies (i) the image of tunder the canonicalhomomorphism from the valuation ring of v onto the residue field of v(henceforth referred to as the v-residue ot t) is transcendental over k<SUB>0</SUB>; that is v coincides with the Gaussian valuation v<SUP>t</SUP><SUB>o</SUB> on the subfield K<SUP>0</SUP>(t) defined by v<SUB>0</SUB><SUP>i</SUP>(∑<SUB>i</SUB>α<SUB>i</SUB>t <SUP>i</SUP>)=min<SUB>i</SUB>(v<SUB>0</SUB>(α<SUB>i</SUB>)); (ii) vis the only valuation of K (up to equivalence) extending the valuation v<SUP>t</SUP><SUB>o