theorem Th82:
  K is having_valuation implies
  NonNegElements(normal-valuation v) = NonNegElements(v)
  proof
    assume
A1: K is having_valuation;
    set f = normal-valuation v;
    thus NonNegElements f c= NonNegElements v
    proof
      let a be object;
      assume a in NonNegElements f;
      then consider x being Element of K such that
A2:   a = x and
A3:   0 <= f.x;
      0 <= v.x by A1,A3,Th41;
      hence thesis by A2;
    end;
    let a be object;
    assume a in NonNegElements v;
    then consider x being Element of K such that
A4: a = x and
A5: 0 <= v.x;
    0 <= f.x by A1,A5,Th41;
    hence thesis by A4;
  end;
