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 Th52:
  K is having_valuation implies
  (0 <= v.a iff a is Element of ValuatRing v)
  proof
    assume K is having_valuation;
    then the carrier of ValuatRing v = NonNegElements v by Def12;
    hence thesis by Th47;
  end;
