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 HM15:
  for F being Field, X being finite-dimensional VectSp of F,
  Y being VectSp of F,
  T being linear-transformation of X, Y
  st T is bijective holds
  Y is finite-dimensional & dim(X) = dim(Y)
  proof
    let F be Field, X be finite-dimensional VectSp of F,
    Y be VectSp of F, T be linear-transformation of X, Y;
    assume AS1: T is bijective;
    hence
    X1: Y is finite-dimensional by HM151;
    for I being Basis of X holds dim(Y) = card I
    proof
      let I be Basis of X;
      dom T = the carrier of X by FUNCT_2:def 1; then
      X12: card I = card (T.:I) by CARD_1:5,AS1,CARD_1:33;
      T.: I is Basis of Y by AS1,HM12;
      hence dim(Y) = card I by X1,X12,VECTSP_9:def 1;
    end;
    hence thesis by VECTSP_9:def 1;
  end;
