reserve a,b,i,j,k,l,m,n for Nat;

theorem SL:
  for f be complex-valued FinSequence, x be Complex holds
    f + x = f + ((len f) |-> x)
  proof
    let f be complex-valued FinSequence, x be Complex;
    reconsider g = ((len f) |-> x) as FinSequence of COMPLEX by NEWTON02:103;
    A0: dom (f+x) = dom f by VALUED_1:def 2;
    A2: len (f+g) = min (len f, len ((len f)|-> x)) by FLS
    .= len f;
    for i be Nat st i in dom (f+x) holds (f+x).i = (f+g).i
    proof
      let i be Nat such that
      B1: i in dom (f+x);
      B2: i in dom f by B1,VALUED_1:def 2;
      reconsider i as non zero Nat by B1,FINSEQ_3:25;
      len f >= i by B2,FINSEQ_3:25; then
      reconsider k = len f - i as Element of NAT by NAT_1:21;
      i in dom (f+g) by A2,B2,FINSEQ_3:29; then
      (f+g).i = f.i + ((k+i)|-> x).i by VALUED_1:def 1
      .= f.i + x;
      hence thesis by B1,VALUED_1:def 2;
    end;
    hence thesis by A0,FINSEQ_3:29,A2,FINSEQ_2:9;
  end;
