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 Th87:
  for V,W be RealLinearSpace,
        A be Subset of V,
        T be LinearOperator of V,W
    st T is bijective
  holds
    A is Basis of V
      iff
    T.:A is Basis of W
  proof
    let V,W be RealLinearSpace;
    let A be Subset of V;
    let T be LinearOperator of V,W;
    assume
    A1: T is bijective;
    reconsider S = T as linear-transformation of RLSp2RVSp(V),RLSp2RVSp(W)
      by Th84;

    reconsider B = A as Subset of RLSp2RVSp(V);
    B is Basis of RLSp2RVSp(V)
      iff
    S .: B is Basis of RLSp2RVSp(W) by VECTSP12:2,A1;

    hence thesis by Th80;
  end;
