
theorem
  for c be XFinSequence of REAL holds
  ex absc be XFinSequence of REAL st
  absc = |. c .| &
  for n be Nat holds (seq_p(c)).n <= (seq_p(absc)).n
  proof
    let c be XFinSequence of REAL;
    reconsider absc = |. c .| as XFinSequence of REAL;
    take absc;
    thus absc = |.c.|;
    let n be Nat;
    CL1: (seq_p(c)).n = Sum(c (#) seq_a^(n,1,0)) by defseqp;
    CL2: (seq_p(absc)).n = Sum(absc (#) seq_a^(n,1,0)) by defseqp;
    set mc = c (#) seq_a^(n,1,0);
    set mac = absc (#) seq_a^(n,1,0);
    px0:dom c =dom absc by VALUED_1:def 11;
    dom mc = dom c /\ dom (seq_a^(n,1,0)) by VALUED_1:def 4; then
    px: len mc = len mac by px0,VALUED_1:def 4;
    for x be Nat st x in dom mc holds mc.x <= mac.x
    proof
      let x be Nat;
      CCL1: mc.x =c.x * (seq_a^(n,1,0)).x by VALUED_1:5;
      CCL2: mac.x =absc.x * (seq_a^(n,1,0)).x by VALUED_1:5;
      PX2:(seq_a^(n,1,0)).x = n to_power (1*x+0) by ASYMPT_1:def 1
      .= n to_power x;
      absc.x = |. c.x .| by VALUED_1:18;
      hence thesis by XREAL_1:64,CCL1,CCL2,ABSVALUE:4,PX2;
    end;
    hence thesis by CL1,CL2,AFINSQ_2:57,px;
  end;
