
theorem LMXFIN5:
  for d be XFinSequence of REAL,k be Nat st len d = 1 holds
  ex a be Real st a = d.0 &
  for x be Nat holds (seq_p(d)).x = a
  proof
    let d be XFinSequence of REAL,k be Nat;
    assume AS: len d = 1;
    reconsider a=d.0 as Real;
    take a;
    thus a=d.0;
    let x be Nat;
    Q1: (seq_p(d)).x = Sum(d (#) seq_a^(x,1,0)) by defseqp;
    Q3:0 in Segm 1 by NAT_1:44;
    Q4:(d (#) seq_a^(x,1,0)).0 = (d.0) * ((seq_a^(x,1,0)).0) by AS,Q3,LMXFIN1
    .= a* x to_power ((1 * 0) + 0) by ASYMPT_1:def 1
    .= a*1 by POWER:24
    .= a;
    len (d (#) seq_a^(x,1,0)) = 1 by AS,LMXFIN1;
    then (d (#) seq_a^(x,1,0)) = <% a %> by AFINSQ_1:34,Q4;
    hence (seq_p(d)).x = a by Q1,AFINSQ_2:53;
  end;
