theorem
  K is having_valuation implies Pgenerator(v) is Element of ValuatRing v
  proof
    set a = Pgenerator(v);
    set l = least-positive(rng v);
    assume
A1: K is having_valuation;
    then
A2: a = the Element of (v"{l}) by Def9;
    l in rng v by A1,Lm5;
    then {l} c= rng v by ZFMISC_1:31;
    then v"{l} is non empty by RELAT_1:139;
    then v.a in {l} by A2,FUNCT_1:def 7;
    then v.a = l by TARSKI:def 1;
    hence thesis by A1,Th52;
  end;
