theorem
  K is having_valuation implies
  (v.a is non negative iff normal-valuation(v).a is non negative)
  proof
    set f = normal-valuation(v);
    set l = least-positive(rng v);
    assume
A1: K is having_valuation;
    then
A2: v.a = f.a*l by Def10;
    per cases;
    suppose v.a is zero or f.a is zero;
      thus thesis by A2;
    end;
    suppose that
A3:   v.a is non zero and
A4:   f.a is non zero;
      thus v.a is non negative implies f.a is non negative by A1,A3,Th40;
      thus thesis by A4,A1,Th40;
    end;
  end;
