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
  for X, Y be RealNormSpace
   st the RLSStruct of X = the RLSStruct of Y
    & X is finite-dimensional
    & dim X <> 0
  holds
    ex k1,k2 be Real
    st 0 < k1
     & 0 < k2
     & for x be Element of X,
           y be Element of Y
         st x = y
       holds
          ||.x.|| <= k1 * ||.y.||
        & ||.y.|| <= k2 * ||.x.||
  proof
    let X, Y be RealNormSpace;
    assume
    A1: the RLSStruct of X = the RLSStruct of Y
      & X is finite-dimensional
      & dim X <> 0;
    set X0 = (Omega).X;
    set Y0 = (Omega).Y;

    A2: X0 = the RLSStruct of X by RLSUB_1:def 4;
    reconsider X0 as finite-dimensional RealLinearSpace by A1;
    A3: dim X = dim X0 by A1,RLVECT_5:30;
    Y0 = the RLSStruct of Y by RLSUB_1:def 4; then
    A4: Y is finite-dimensional by A1,RLAFFIN3:3,A2;
    reconsider Y0 as finite-dimensional RealLinearSpace
      by A1,A2,RLSUB_1:def 4;

    X0 = Y0 by A1,A2,RLSUB_1:def 4; then

    A6: dim X = dim Y by A3,A4,RLVECT_5:30;

    set b = the OrdBasis of RLSp2RVSp(X0);
    reconsider e = b as OrdBasis of RLSp2RVSp(Y0) by A1,A2,RLSUB_1:def 4;

    consider k1,k2 be Real such that
    A7: 0 < k1 & 0 < k2
      & for x be Point of X
        holds
          ||.x.|| <= k1 * max_norm(X0,b).x
        & max_norm(X0,b).x <= k2 * ||.x.|| by A1,A2,Th25;

    Y0 = the RLSStruct of Y by RLSUB_1:def 4; then
    consider j1,j2 be Real such that
    A8: 0 < j1 & 0 < j2
      & for x be Point of Y
        holds
          ||.x.|| <= j1 * max_norm(Y0,e).x
        & max_norm(Y0,e).x <= j2 * ||.x.|| by A1,A4,A6,Th25;

    A9:
    now let x be Element of X,
      y be Element of Y;
      assume x = y; then

      A12: max_norm(X0,b).x = max_norm(Y0,e).y by A1,A2,RLSUB_1:def 4;

      A11: ||.x.|| <= k1 * max_norm(X0,b).x
         & max_norm(X0,b).x <= k2 * ||.x.||
         & ||.y.|| <= j1 * max_norm(Y0,e).y
         & max_norm(Y0,e).y <= j2 * ||.y.|| by A7,A8;

      k1 * max_norm(Y0,e).y <= k1 * (j2 * ||.y.||) by A7,A8,XREAL_1:64;
      hence ||.x.|| <= k1 * j2 * ||.y.|| by A11,A12,XXREAL_0:2;

      j1 * max_norm(X0,b).x <= j1 * (k2 * ||.x.||) by A7,A8,XREAL_1:64;
      hence ||.y.|| <= j1 * k2 * ||.x.|| by A11,A12,XXREAL_0:2;
    end;
    set h1 = k1 * j2;
    set h2 = j1 * k2;
    take h1,h2;
    thus 0 < h1 & 0 < h2 by A7,A8,XREAL_1:129;
    thus thesis by A9;
  end;
