theorem
  K is having_valuation & a <> 0.K &
  a is Element of ValuatRing v & a" is Element of ValuatRing v implies
  v.a = 0
  proof
    assume
A1: K is having_valuation;
    assume that
A2: a <> 0.K and
A3: a is Element of ValuatRing v;
    assume a" is Element of ValuatRing v;
    then 0 <= v.a" by A1,Th52;
    then --v.a <= -0 by A1,A2,Th21;
    hence thesis by A3,A1,Th52;
  end;
