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 Th27:
  for V be RealNormSpace
   st V is finite-dimensional & dim V <> 0
  holds
    ex k1,k2 be Real,
       S be LinearOperator of V, REAL-NS(dim V)
    st S is bijective
     & 0 <= k1
     & 0 <= k2
     & for x be Element of V
       holds
          ||.S.x.|| <= k1 * ||.x.||
        & ||.x.|| <= k2 * ||.S.x.||
  proof
    let V be RealNormSpace;
    assume
    A1: V is finite-dimensional & dim V <> 0;

    then
    consider S be LinearOperator of V, REAL-NS(dim V),
             W be finite-dimensional VectSp of F_Real,
             b be OrdBasis of W such that
    A2: W = RLSp2RVSp(V)
      & S is bijective
      & for x be Element of W
        holds S.x = x |-- b by Th24;

    reconsider V0 = V as finite-dimensional RealLinearSpace by A1;
    reconsider b0 = b as OrdBasis of RLSp2RVSp(V0) by A2;
    the RLSStruct of V0 = the RLSStruct of V; then

    consider k1,k2 be Real such that
    A4: 0 < k1 & 0 < k2
      & for x be Point of V
        holds ||.x.|| <= k1 * max_norm(V0,b0).x
            & max_norm(V0,b0).x <= k2 * ||.x.|| by Th25,A1;

    A5:
    now
      let x be Element of V;
      reconsider Sx = S.x as Element of REAL dim(V) by REAL_NS1:def 4;

      consider x1 be Element of RLSp2RVSp(V0),
               z1 be Element of REAL(dim V0) such that
      A6: x = x1
        & z1 = x1 |-- b0
        & (euclid_norm(V0,b0)).x = |.z1.| by Def5;

      A8: (euclid_norm(V0,b0)).x = ||.S.x.|| by A2,A6,REAL_NS1:1;

      (sum_norm(V0,b0)).x <= (dim V) * (max_norm(V0,b0)).x
        &
      (max_norm(V0,b0)).x <= (euclid_norm(V0,b0)).x
        &
      (euclid_norm(V0,b0)).x <= (sum_norm(V0,b0)).x by Th22,A1;

      then
      A9: ||.S.x.|| <= (dim V0) * (max_norm(V0,b0)).x by A8,XXREAL_0:2;

      (dim V0) * max_norm(V0,b0).x <= (dim V0) * (k2 * ||.x.||)
        by A4,XREAL_1:64;
      hence ||.S.x.|| <= (dim V0) * k2 * ||.x.|| by A9,XXREAL_0:2;

      A10: k1 * (max_norm(V0,b0)).x <= k1 * ||.S.x.||
        by A1,A4,A8,Th22,XREAL_1:64;

      ||.x.|| <= k1 * max_norm(V0,b0).x by A4;
      hence ||.x.|| <= k1 * ||.S.x.|| by A10,XXREAL_0:2;
    end;

    set h1 = (dim V0) * k2;
    take h1,k1,S;
    thus S is bijective by A2;
    thus 0 <= h1 & 0 <= k1 by A4;
    thus thesis by A5;
  end;
