
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 A1,A2 be Vector of VectQuot(V,W), v1,v2 be Vector of
  V st A1 = v1 + W & A2 = v2 + W holds A1 + A2 = v1 + v2 + 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);
  let A1,A2 be Vector of vw, v1,v2 be Vector of V;
  assume
A1: A1 = v1 + W & A2 = v2 + W;
  A2 is Coset of W by Th22;
  then
A2: A2 in the set of all B where B is Coset of W;
  A1 is Coset of W by Th22;
  then A1 in the set of all B where B is Coset of W;
  then reconsider w1=A1,w2=A2 as Element of CosetSet(V,W) by A2;
  thus A1+A2 = (addCoset(V,W)).(w1,w2) by Def6
    .= (v1 + v2) + W by A1,Def3;
end;
