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 Th8:
  for n be non empty Nat,
      x be Element of REAL n
  holds
    ex xMAX be Real
    st xMAX in rng(abs x)
      &
    for i be Nat
     st i in dom x
    holds abs(x).i <= xMAX
  proof
    let n be non empty Nat,
        x be Element of REAL n;
    set F = rng(abs x);
    A1: F is bounded_above
      & upper_bound F in F by SEQ_4:133;

    set xMAX = upper_bound F;
    reconsider xMAX as Real;
    take xMAX;
    thus xMAX in rng(abs x) by SEQ_4:133;

    thus
    for i be Nat st i in dom x
    holds abs(x).i <= xMAX
    proof
        let i be Nat;
        assume i in dom x;
        then i in dom(abs x) by VALUED_1:def 11;
        then abs(x).i in F by FUNCT_1:3;
        hence abs(x).i <= xMAX by A1,SEQ_4:def 1;
    end;
  end;
