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

theorem SUM:
  for f be complex-valued FinSequence holds
    Sum f = Sum (f|i) + f.(i+1) + Sum (f/^(i+1))
  proof
    let f be complex-valued FinSequence;
   set f1 = f|i, f2 = f/^i, k = 1,f4 = f2/^k;
   Sum f = Sum (f1^f2) by RFINSEQ:8
    .= Sum (f|i) + Sum f2 by RVSUM_2:32
    .= Sum (f|i) + Sum ((f2|k)^(f2/^k)) by RFINSEQ:8
    .= Sum (f|i) + (Sum (f2|k) + Sum (f2/^k)) by  RVSUM_2:32
    .= Sum (f|i) + (Sum (f2|k) + Sum (f/^(i+k))) by FINSEQ681
    .= Sum (f|i) + Sum (f2|k) + Sum (f/^(i+k));
    hence thesis by S2;
  end;
