
theorem
  for f be Real_Sequence st ex n be Nat st (for k be Nat st k >= n holds f.k=0)
  holds ex n be Nat st for k be Nat st k >= n holds
  (Partial_Sums f).k = (Partial_Sums f).n
  proof
    let f be Real_Sequence;
    given n being Nat such that
A1: for k being Nat st k >= n holds f.k = 0;
    set p = Partial_Sums f;
    reconsider pk = p.n as Real;
    set r = seq_const pk;
A2: for k being Nat st k >= n holds p.k = r.k
    proof
      let k be Nat;
      assume
A3:   k >= n;
      defpred P[Nat] means p.$1 = r.$1;
A4:   P[n] by SEQ_1:57;
A5:   for i being Nat st n <= i holds P[i] implies P[i+1]
        proof
         let i be Nat;
         assume
A6:      n <= i;
         assume
A7:      P[i];
         p.(i+1)
       = p.i + f.(i+1) by SERIES_1:def 1
      .= r.i + 0 by A1,A6,A7,NAT_1:12
      .= pk by SEQ_1:57
      .= r.(i + 1);
        hence thesis;
      end;
      for k being Nat st n <= k holds P[k] from NAT_1:sch 8(A4,A5);
      hence thesis by A3;
    end;
    take n;
    let k be Nat;
    assume k >= n; then
    p.k = r.k by A2 .= p.n by SEQ_1:57;
    hence thesis;
  end;
