theorem Th72:
  K is having_valuation & v.a = 0 implies
  a is Element of ValuatRing v & a" is Element of ValuatRing v
  proof
    assume
A1: K is having_valuation;
    assume
A2: v.a = 0;
    thus a is Element of ValuatRing v by A1,A2,Th52;
    a <> 0.K by A1,A2,Def8;
    then v.a" = -v.a by A1,Th21;
    hence thesis by A1,A2,Th52;
  end;
