
theorem LMXFIN17:
  for c be XFinSequence of REAL,
  d be XFinSequence of REAL
  st len d = len c &
  for i be Nat st i in dom d holds d.i = |. c.i .| holds
  for n be Nat holds ( seq_p(c) ).n <= ( seq_p(d) ).n
  proof
    let c be XFinSequence of REAL,
    d be XFinSequence of REAL;
    assume AS:len d = len c &
    for i be Nat st i in dom d holds d.i = |. c.i .|;
    let x be Nat;
    P1: (seq_p(c)).x = Sum(c (#) seq_a^(x,1,0)) by defseqp;
    P2: (seq_p(d)).x = Sum(d (#) seq_a^(x,1,0)) by defseqp;
    dom (d (#) seq_a^(x,1,0)) = dom d by LMXFIN1
    .= dom (c (#) seq_a^(x,1,0)) by LMXFIN1,AS; then
    D1: len (d (#) seq_a^(x,1,0)) = len (c (#) seq_a^(x,1,0));
    for i be Nat st i in dom (c (#) seq_a^(x,1,0)) holds
    (c (#) seq_a^(x,1,0)).i <= (d (#) seq_a^(x,1,0)).i
    proof
      let i be Nat;
      assume i in dom (c (#) seq_a^(x,1,0)); then
      P6:i in dom c by LMXFIN1; then
      P7: (c (#) seq_a^(x,1,0)).i = (c.i) * x to_power (i) by LMXFIN2;
      P8: (d (#) seq_a^(x,1,0)).i = (d.i) * x to_power (i) by P6,AS,LMXFIN2;
      d.i = |. c.i .| by AS,P6;
      hence thesis by XREAL_1:64,P7,P8,ABSVALUE:4;
    end;
    hence ( seq_p(c) ).x <= ( seq_p(d) ).x by D1,AFINSQ_2:57,P1,P2;
  end;
