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 Th14:
  for n be non empty Nat,
      x be Element of REAL n
  holds
      (sum_norm(n)).x <= n * (max_norm(n)).x
    & (max_norm(n)).x <= |.x.|
    & |.x.| <= (sum_norm(n)).x
  proof
    let n be non empty Nat,
        x be Element of REAL n;
    set xMAX = (max_norm(n)).x;

    xMAX in rng (abs x)
      &
    for i be Nat st i in dom x
    holds abs(x).i <= xMAX by Def1;

    then
    Sum (abs x) <= n* xMAX
      &
    xMAX <= |.x.|
      &
    |.x.| <= Sum (abs x) by Th11;
    hence thesis by Def2;
  end;
