
theorem Th23:
  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, v be Vector of V holds v+W is Coset of W & v+W
  is Vector of VectQuot(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, v be Vector of V;
  set cs = CosetSet(V,W);
  thus v + W is Coset of W by VECTSP_4:def 6;
  then v+W in cs;
  hence thesis by Def6;
end;
