
theorem TLNEG41:
  for c,absc be XFinSequence of REAL st absc = |. c .|
  holds for n be Nat holds |.(seq_p(c)).n .| <= (seq_p(absc)).n
  proof
    defpred P[Nat] means for c,absc be XFinSequence of REAL
    st len c = $1 & absc = |. c .|
    holds for x be Nat holds |.(seq_p(c)).x .| <= (seq_p(absc)).x;
    P0:P[0]
    proof
      let c,absc be XFinSequence of REAL;
      assume A0: len c = 0 & absc = |. c .|;
      D2: dom absc = {} by A0,VALUED_1:def 11;
      let x be Nat;
      c (#) seq_a^(x,1,0) = {} by A0,LMXFIN1; then
      Q2: Sum(c (#) seq_a^(x,1,0)) = 0;
      absc (#) seq_a^(x,1,0) = {} by D2,LMXFIN1; then
      Q3: Sum(absc (#) seq_a^(x,1,0)) = 0;
      |. (seq_p(c)).x .| = 0 by COMPLEX1:44,Q2,defseqp;
      hence |.(seq_p(c)).x .| <= (seq_p(absc)).x by Q3,defseqp;
    end;
    P1:for k be Nat st P[k] holds P[k+1]
    proof
      let k be Nat;
      assume A0: P[k];
      let c,absc be XFinSequence of REAL;
      assume A1: len c = k+1 & absc = |. c .|;
      consider a1 be Real,c1 be XFinSequence of REAL,
      y1 be Real_Sequence such that
      A2: len c1 = k & c1= c | k & a1 = c.k & c =c1^<% a1 %> &
      seq_p(c) = seq_p(c1) + y1 &
      for i be Nat holds
      y1.i = a1* (i to_power k) by A1,LMXFIN4;
      len absc = k+1 by A1,VALUED_1:def 11; then
      consider a2 be Real,c2 be XFinSequence of REAL,
      y2 be Real_Sequence such that
      A4: len c2 = k & c2= absc | k & a2 = absc.k & absc =c2^<% a2 %> &
      seq_p(absc) = seq_p(c2) + y2 &
      for i be Nat holds y2.i = a2* (i to_power k) by LMXFIN4;
      A5: |. a1 .| = a2 by A1,A2,A4,VALUED_1:18;
      for i being object st i in dom c2 holds c2.i = |. c1.i .|
      proof
        let i be object;
        assume B1: i in dom c2;
        c2.i = absc.i by A4,B1,FUNCT_1:47
        .= |. c.i .| by A1,VALUED_1:18
        .= |. c1.i .| by A2,B1,FUNCT_1:47,A4;
        hence thesis;
      end; then
      AA7:c2= |. c1 .| by VALUED_1:def 11,A2,A4;
      let x be Nat;
      A8: (seq_p(c)).x = (seq_p(c1)).x + y1.x by SEQ_1:7,A2;
      A9: (seq_p(absc)).x = (seq_p(c2)).x + y2.x by SEQ_1:7,A4;
      A10: |.(seq_p(c)).x .| <= |.(seq_p(c1)).x .| + |.y1.x .|
      by A8,COMPLEX1:56;
      A11: |.(seq_p(c1)).x .| <= (seq_p(c2)).x by AA7,A2,A0;
      y1.x = a1* (x to_power k) by A2; then
      |.y1.x .| = |. a1 .| * |. x to_power k .| by COMPLEX1:65
      .=|. a1 .| * (x to_power k) by ABSVALUE:def 1
      .= y2.x by A4,A5; then
      |.(seq_p(c1)).x .| + |.y1.x .| <= (seq_p(c2)).x + y2.x
      by XREAL_1:7,A11;
      hence |.(seq_p(c)).x .| <= (seq_p(absc)).x by A9,XXREAL_0:2,A10;
    end;
    X1: for n be Nat holds P[n] from NAT_1:sch 2(P0,P1);
    let c,absc be XFinSequence of REAL;
    assume X2: absc = |. c .|;
    len c is Nat;
    hence for n be Nat holds |.(seq_p(c)).n .| <= (seq_p(absc)).n by X1,X2;
  end;
