theorem Th77:
  K is having_valuation implies
  for x being Element of ValuatRing v st not x in vp(v)
  holds v.x = 0
  proof
    assume
A1: K is having_valuation;
    let x be Element of ValuatRing v;
    reconsider y = x as Element of K by A1,Th51;
    assume not x in vp(v);
    then v.y <= 0 by A1,Th61;
    hence thesis by A1,Th52;
  end;
