
theorem Th4:
  for rseq be Real_Sequence st rseq is absolutely_summable & Sum
  abs rseq=0 holds for n be Nat holds rseq.n =0
proof
  let rseq being Real_Sequence such that
A1: rseq is absolutely_summable and
A2: Sum abs rseq=0;
  reconsider arseq = abs rseq as Real_Sequence;
A3: for n be Nat holds 0 <= (abs rseq).n
  proof
    let n be Nat;
    0 <= |.rseq.n.| by COMPLEX1:46;
    hence thesis by SEQ_1:12;
  end;
A4: arseq is summable by A1,SERIES_1:def 4;
    let n be Nat;
    (abs rseq).n =0 by A2,A4,A3,RSSPACE:17;
    then |.rseq.n.|=0 by SEQ_1:12;
    hence thesis by ABSVALUE:2;
end;
