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 Th3:
  for x be Element of REAL(n+1),
      y be Element of REAL n,
      w be Element of REAL
    st y = x | n & w = x.(n+1)
  holds |.x.| <= |.y.| + |.w.|
  proof
    let x be Element of REAL (n+1),
        y be Element of REAL n,
        w be Element of REAL;
    assume
    A1: y = x | n & w = x.(n+1);
    len x = n+1 by CARD_1:def 7; then
    x = y ^ <*w*> by A1,FINSEQ_3:55; then

    sqr x = (sqr y) ^ <* sqrreal.w *> by FINSEQOP:8;
    then
    A3: Sum (sqr x) = Sum (sqr y) + sqrreal.w by RVSUM_1:74;
    A4: 0 <= Sum (sqr y) by RVSUM_1:86;
    A5: sqrreal.w = w ^2 by RVSUM_1:def 2;
    then 0 <= sqrreal.w by XREAL_1:63;
    then sqrt(Sum (sqr x))
      <= sqrt(Sum (sqr y)) + sqrt (sqrreal.w) by A3,A4,SQUARE_1:59;
    hence thesis by A5,COMPLEX1:72;
  end;
