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 Th89:
  for V be finite-dimensional RealLinearSpace
    st dim V <> 0
  ex T be LinearOperator of V, RealVectSpace(Seg(dim V))
    st T is bijective
  proof
    let V be finite-dimensional RealLinearSpace;
    assume
    A1: dim V <> 0;

    RLSp2RVSp(V) is finite-dimensional
      &
    dim RLSp2RVSp(V) = dim V by Th81; then
    consider T be linear-transformation
      of RLSp2RVSp(V), (dim V) -VectSp_over F_Real
    such that
    A2: T is bijective by Th64;

    (dim V) -VectSp_over F_Real
    = RLSp2RVSp(RealVectSpace(Seg(dim V))) by Th83,A1;
    then
    reconsider S = T as LinearOperator of V,RealVectSpace(Seg(dim V)) by Th84;
    take S;
    the carrier of RealVectSpace(Seg(dim V))
     = the carrier of (RLSp2RVSp(RealVectSpace(Seg(dim V))))
    .= the carrier of ((dim V) -VectSp_over F_Real) by Th83,A1;
    hence S is bijective by A2;
  end;
