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 Th24:
  for V be finite-dimensional RealNormSpace
    st dim V <> 0
  holds
    ex S be LinearOperator of V, REAL-NS(dim V),
       W be finite-dimensional VectSp of F_Real,
       b be OrdBasis of W
    st W = RLSp2RVSp(V)
     & S is bijective
     & for x be Element of W
       holds S.x = x |-- b
  proof
    let V be finite-dimensional RealNormSpace;
    assume
    A1: dim V <> 0;
    A2: RLSp2RVSp(V) is finite-dimensional
      & dim RLSp2RVSp(V) = dim V by REAL_NS2:81;

    reconsider W = RLSp2RVSp (V) as finite-dimensional VectSp of F_Real;
    set b = the OrdBasis of W;

    consider T be linear-transformation of W,(dim V) -VectSp_over F_Real
    such that
    A3: T is bijective
      & for x be Element of W
        holds T.x = x |-- b by A2,REAL_NS2:63;

    (dim V) -VectSp_over F_Real
    = RLSp2RVSp(RealVectSpace(Seg(dim V))) by A1,REAL_NS2:83;
    then
    reconsider S = T as LinearOperator of V,RealVectSpace(Seg(dim V))
      by REAL_NS2:84;

    A4: 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 A1,REAL_NS2:83;

    A5: the RLSStruct of REAL-NS(dim V)
      = the RLSStruct of TOP-REAL(dim V) by REAL_NS2:1
     .= RealVectSpace(Seg(dim V)) by EUCLID:def 8;
    then
    reconsider K = S as Function of the carrier of V,
                                    the carrier of REAL-NS(dim V);

    for x, y be Element of V
    holds K.(x + y) = K.x + K.y
    proof
      let x, y be Element of V;
      S is additive;
      hence K.(x + y)
       = S.x + S.y
      .= K.x + K.y by Th16;
    end; then
    A7: K is additive;

    for a be Real,
        x be VECTOR of V
    holds K.(a * x) = a * (K . x)
    proof
      let a be Real;
      let x be VECTOR of V;
      thus K.(a * x)
       = a * S.x by LOPBAN_1:def 5
      .= a * K.x by Th16;
    end;
    then K is homogeneous;
    then reconsider K as LinearOperator of V,REAL-NS (dim V) by A7;
    take K,W,b;
    thus thesis by A3,A4,A5;
  end;
