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 Th47:
  for K being non empty doubleLoopStr,
      v being Valuation of K,
      a being Element of K holds
  a in NonNegElements v iff 0 <= v.a
  proof
    let K be non empty doubleLoopStr,
    v be Valuation of K,
    a be Element of K;
    hereby
      assume a in NonNegElements v;
      then ex x being Element of K st a = x & 0 <= v.x;
      hence 0 <= v.a;
    end;
    thus thesis;
  end;
