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 HM12:
  for F being Field, X, Y being VectSp of F,
  T being linear-transformation of X, Y,
  A being Subset of X st T is bijective holds
  A is Basis of X iff T.: A is Basis of Y
  proof
    let F be Field, X, Y be VectSp of F,
    L be linear-transformation of X, Y,
    A be Subset of X;
    assume AS1: L is bijective;
    D1: dom L = the carrier of X by FUNCT_2:def 1;
    consider K be linear-transformation of Y, X such that
    AS3: K= L" & K is bijective by ZMODUL06:42,AS1;
    thus A is Basis of X implies L.:A is Basis of Y by AS1,HM11;
    assume L.:A is Basis of Y;
    then K.: (L.:A) is Basis of X by AS3,HM11;
    hence thesis by D1,AS1,AS3,FUNCT_1:107;
  end;
