
theorem Th16:
  for f be INT-valued Real_Sequence st ex n be Nat
  st (for k be Nat st k >= n holds f.k = 0) holds Sum f is Integer
  proof
    let f be INT-valued Real_Sequence;
    given n be Nat such that
A1: for k be Nat st k >= n holds f.k = 0;
    set p = Partial_Sums f;
    reconsider pk = p.n as Real;
    set r = seq_const pk;
    for k be Nat st k >= n holds p.k = r.k
    proof
      let k be Nat;
      assume
A2:   k >= n;
      defpred P[Nat] means p.$1 = r.$1;
A3:   P[n] by SEQ_1:57;
A4:   for i be Nat st n <= i holds P[i] implies P[i+1]
      proof
        let i be Nat;
        assume
A5:     n <= i;
        assume
A6:     P[i];
        p.(i+1)
      = p.i + f.(i+1) by SERIES_1:def 1 .= r.i + 0 by A1,A5,A6,NAT_1:12
     .= pk by SEQ_1:57 .= r.(i+1);
        hence thesis;
      end;
      for k be Nat st n <= k holds P[k] from NAT_1:sch 8(A3,A4);
      hence thesis by A2;
    end; then
    lim p = lim r by SEQ_4:19;
    hence thesis;
  end;
