reserve K,F for Ring;
reserve V,W for VectSp of K;
reserve l for Linear_Combination of V;
reserve T for linear-transformation of V,W;

theorem
  for K being Field
  for V being finite-dimensional VectSp of K, W being Subspace of V holds
  VectQuot(V,W) is finite-dimensional &
  dim VectQuot(V,W) + dim W = dim V
  proof
    let K be Field;
    let V be finite-dimensional VectSp of K,
    W be Subspace of V;
    set Vq = VectQuot(V,W);
    consider S be Linear_Compl of W,
    T be linear-transformation of S, Vq such that
    X1: T is bijective and
    for v being Vector of V st v in S holds T.v = v+ W by Th38A;
    Vq is finite-dimensional & dim(S) = dim(Vq) by HM15,X1;
    hence thesis by VECTSP_5:def 5,VECTSP_9:34;
  end;
