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;
