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 Th2:
  for x be Element of REAL(n+1),
      w be Element of REAL
  st w = x.(n+1)
  holds |.w.| <= |.x.|
  proof
    let x be Element of REAL(n+1);
    let w be Element of REAL;
    assume
    A1: w = x.(n+1);
    reconsider y = x as Point of TOP-REAL(n+1) by EUCLID:22;
    reconsider y as FinSequence of REAL;

    A2: len x = n+1 by CARD_1:def 7;
    A3: n+1 in Seg (n+1) by FINSEQ_1:4;
    then
    A4: |.(y /. (n+1)).| <= |.x.| by PDIFF_8:1;
    n + 1 in dom y by A2,FINSEQ_1:def 3,A3;
    hence thesis by PARTFUN1:def 6,A4,A1;
  end;
