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 Th1:
  for x be Element of REAL(n + 1),
      y be Element of REAL n
    st y = x | n
  holds |.y.| <= |.x.|
  proof
    let x be Element of REAL(n+1),
        y be Element of REAL n;
    assume
    A1: y = x | n;

    A2: len x = n + 1 by CARD_1:def 7;
    n + 1 in Seg(n + 1) by FINSEQ_1:4;
    then n + 1 in dom x by A2,FINSEQ_1:def 3;
    then x.(n + 1) in rng x by FUNCT_1:3;
    then reconsider xn = x.(n+1) as Element of REAL;

    x = y^<*xn*> by A1,A2,FINSEQ_3:55; then
    sqr x = (sqr y) ^ <* sqrreal.xn *> by FINSEQOP:8;

    then
    A4: Sum(sqr x) = Sum(sqr y) + sqrreal.xn by RVSUM_1:74;
    A5: 0 <= Sum(sqr y) by RVSUM_1:86;
    sqrreal.xn = xn ^2 by RVSUM_1:def 2;
    then 0 <= sqrreal.xn by XREAL_1:63;
    then 0 + Sum (sqr y) <= Sum (sqr x) by A4,XREAL_1:7;
    hence thesis by A5,SQUARE_1:26;
  end;
