theorem Th48:
  for K being non empty doubleLoopStr,
      v being Valuation of K holds
  NonNegElements v c= the carrier of K
  proof
    let K be non empty doubleLoopStr,
    v be Valuation of K;
    let a be object;
    assume a in NonNegElements v;
    then ex x being Element of K st a = x & 0 <= v.x;
    hence thesis;
  end;
