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 HM151:
  for F being Field, X, Y being VectSp of F,
  T being linear-transformation of X, Y
  st T is bijective holds
  X is finite-dimensional iff Y is finite-dimensional
  proof
    let F be Field, X, Y be VectSp of F,
    T be linear-transformation of X, Y;
    assume AS1: T is bijective;
    consider K be linear-transformation of Y, X such that
    AS3: K= T" & K is bijective by ZMODUL06:42,AS1;
    thus thesis by HM13,AS1,AS3;
  end;
