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 Th22:
  for X be finite-dimensional RealLinearSpace,
      b be OrdBasis of RLSp2RVSp (X),
      x be Element of X
    st dim X <> 0
  holds
      (sum_norm(X,b)).x <= (dim X)* (max_norm(X,b)).x
    & (max_norm(X,b)).x <= (euclid_norm(X,b)).x
    & (euclid_norm(X,b)).x <= (sum_norm(X,b)).x
  proof
    let X be finite-dimensional RealLinearSpace,
        b be OrdBasis of RLSp2RVSp (X),
        x be Element of X;
    assume
    A1: dim X <> 0;

    consider x1 be Element of RLSp2RVSp (X),
             z1 be Element of REAL (dim X) such that
    A2: x = x1
      & z1 = (x1 |-- b)
      & (max_norm(X,b)).x = (max_norm(dim X)).z1 by Def3;

    consider x2 be Element of RLSp2RVSp (X),
             z2 be Element of REAL (dim X) such that
    A3: x = x2
     & z2 = (x2 |-- b)
     & (sum_norm(X,b)).x = (sum_norm(dim X)).z2 by Def4;

    consider x3 be Element of RLSp2RVSp (X),
             z3 be Element of REAL dim X such that
    A4: x = x3
      & z3 = (x3 |-- b)
      & (euclid_norm(X,b)).x = |.z3.| by Def5;
    thus thesis by A1,A2,A3,A4,Th14;
  end;
