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
  K is having_valuation implies
  (v.a = v.b iff normal-valuation(v).a = normal-valuation(v).b)
  proof
    set f = normal-valuation(v);
    set l = least-positive(rng v);
    assume
A1: K is having_valuation;
    then
A2: l in REAL by Lm6;
    v.a = (f.a)*l & v.b = (f.b)*l by A1,Def10;
    hence thesis by A2,XXREAL_3:68;
  end;
