theorem Th51:
  K is having_valuation implies
  for x being Element of ValuatRing v holds x is Element of K
  proof
    assume
A1: K is having_valuation;
    let x be Element of ValuatRing v;
    the carrier of ValuatRing v = NonNegElements v by A1,Def12; then
    x in NonNegElements v & NonNegElements v c= the carrier of K by Th48;
    hence thesis;
  end;
