reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem Th88:
  for V be finite-dimensional RealLinearSpace,
      W be RealLinearSpace
    st (ex T be LinearOperator of V,W
        st T is bijective)
  holds
    W is finite-dimensional
      &
    dim W = dim V
  proof
    let V be finite-dimensional RealLinearSpace,
        W be RealLinearSpace;
    given T be LinearOperator of V,W such that
    A1: T is bijective;
    consider A be finite Subset of V such that
    A2: A is Basis of V by RLVECT_5:def 1;
    A3: dom T = the carrier of V by FUNCT_2:def 1;
    A4: T.:A is Basis of W by A1,A2,Th87;
    hence W is finite-dimensional by RLVECT_5:def 1;
    hence dim W
     = card (T.:A) by A4,RLVECT_5:def 2
    .= card A by A1,A3,CARD_1:5,CARD_1:33
    .= dim V by A2,RLVECT_5:def 2;
  end;
