
theorem Th22:
  for K be add-associative right_zeroed right_complementable
Abelian associative well-unital distributive non empty doubleLoopStr for V be
VectSp of K, W be Subspace of V, w be Vector of VectQuot(V,W) holds w is Coset
  of W & ex v be Vector of V st w = v + W
proof
  let K be add-associative right_zeroed right_complementable Abelian
associative well-unital distributive non empty doubleLoopStr, V be VectSp of
  K, W be Subspace of V, w be Vector of VectQuot(V,W);
  set qv = VectQuot(V,W), cs = CosetSet(V,W);
  cs = the carrier of qv by Def6;
  then w in the set of all A where A is Coset of W;
  then ex A be Coset of W st A= w;
  hence thesis by VECTSP_4:def 6;
end;
