reserve x, y, z, s for ExtReal;
reserve i, j for Integer;
reserve n, m for Nat;
reserve x, y, v, u for ExtInt;
reserve
  D for non empty doubleLoopStr,
  A for Subset of D;
reserve K for Field-like non degenerated
  associative add-associative right_zeroed right_complementable
  distributive Abelian non empty doubleLoopStr,
  a, b, c for Element of K;
reserve v for Valuation of K;

theorem Th40:
  K is having_valuation implies
  (v.a is positive iff normal-valuation(v).a is positive)
  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;
    reconsider l1 = l as Element of REAL by A1,Lm6;
    hereby
      assume
A3:   v.a is positive;
      per cases by A3,XXREAL_3:1;
      suppose v.a is positive Real;
        then reconsider va = v.a as positive Real;
A4:     va in REAL by XREAL_0:def 1;
        then f.a in REAL by A2,XXREAL_3:73;
        then consider c, b being Complex such that
A5:     f.a = c & l1 = b & f.a*l1 = c*b by XXREAL_3:def 5;
        reconsider c as Element of REAL by A4,A5,A2,XXREAL_3:73;
        va = c*b by A1,Def10,A5;
        hence f.a is positive by A5;
      end;
      suppose v.a = +infty;
        hence f.a is positive by A1,Th38;
      end;
    end;
    assume
A6: f.a is positive;
    per cases by A6,XXREAL_3:1;
    suppose f.a is positive Real;
      then reconsider fa = f.a as positive Real;
      v.a = fa*l1 by A2,XXREAL_3:def 5;
      hence v.a is positive;
    end;
    suppose f.a = +infty;
      hence v.a is positive by A1,Th38;
    end;
  end;
