
theorem
  for 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 for A be Vector of VectQuot(V,W), v be Vector of V, a be
  Scalar of V st A = v + W holds a*A = a*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;
  set vw = VectQuot(V,W), lm = the lmult of vw;
  let A be Vector of vw, v be Vector of V, a be Scalar of V;
  assume
A1: A = v + W;
  A is Coset of W by Th22;
  then A in the set of all B where B is Coset of W;
  then reconsider w=A as Element of CosetSet(V,W);
  thus a*A = lm.(a,A)
    .= (lmultCoset(V,W)).(a,w) by Def6
    .= a*v + W by A1,Def5;
end;
