
theorem
  for rseq be Real_Sequence st (for n be Nat holds 0 <= rseq.n) &
  rseq is summable & Sum rseq=0 holds for n be Nat holds rseq.n = 0
proof
  let rseq be Real_Sequence such that
A1: for n be Nat holds 0 <= rseq.n and
A2: rseq is summable and
A3: Sum rseq=0;
A4: Partial_Sums rseq is bounded_above by A1,A2,SERIES_1:17;
A5: for n be Nat holds (Partial_Sums rseq).n <= Sum rseq
  proof
    let n be Nat;
    (Partial_Sums(rseq)).n <= lim Partial_Sums rseq by A1,A4,SEQ_4:37
,SERIES_1:16;
    hence thesis by SERIES_1:def 3;
  end;
A6: Partial_Sums rseq is non-decreasing by A1,SERIES_1:16;
  now
    given n1 be Nat such that
A7: rseq.n1 <> 0;
A8: for n be Nat holds 0 <= Partial_Sums(rseq).n
    proof
      let n be Nat;
A9:   n=n+0 & Partial_Sums(rseq).0 = rseq.0 by SERIES_1:def 1;
      0 <=rseq.0 by A1;
      hence thesis by A6,A9,SEQM_3:5;
    end;
    Partial_Sums(rseq).n1 >0
    proof
      now
        per cases;
        case
A10:      n1=0;
          then Partial_Sums(rseq).n1=rseq.0 by SERIES_1:def 1;
          hence thesis by A1,A7,A10;
        end;
        case
A11:      n1<>0;
          set nn=n1-1;
A12:      nn+1 =n1 & 0 <= rseq.n1 by A1;
A13:       n1 in NAT by ORDINAL1:def 12;
          0 <= n1 by NAT_1:2;
          then 0 + 1 <= n1 by A11,INT_1:7,A13;
          then
A14:      nn in NAT by INT_1:5,A13;
          then
A15:       Partial_Sums(rseq).(nn+1) = Partial_Sums(rseq).nn + rseq.(nn+1)
          by SERIES_1:def 1;
          0 <= Partial_Sums(rseq).nn by A8,A14;
          hence thesis by A7,A12,A15;
        end;
      end;
      hence thesis;
    end;
    hence contradiction by A3,A5;
  end;
  hence thesis;
end;
