
theorem TLNEG1:
  for c be XFinSequence of REAL st len c = 0 holds
  for x be Nat holds (seq_p(c)).x = 0
  proof
    let c be XFinSequence of REAL;
    assume AS:len c = 0;
    let x be Nat;
    L1:(seq_p(c)).x = Sum(c (#) seq_a^(x,1,0)) by defseqp;
    reconsider f =c (#) seq_a^(x,1,0) as Sequence;
    dom f = (dom c) /\ (dom seq_a^(x,1,0)) by VALUED_1:def 4;
    then L2: f = {} by AS;
    reconsider f as XFinSequence of REAL;
    thus thesis by L2,L1;
  end;
