reserve rseq, rseq1, rseq2 for Real_Sequence;
reserve seq, seq1, seq2 for Complex_Sequence;
reserve k, n, n1, n2, m for Nat;
reserve p, r for Real;
reserve z for Complex;
reserve Nseq,Nseq1 for increasing sequence of NAT;

theorem Th57:
  Re seq is summable & Im seq is summable implies seq is summable
  & Sum(seq)=Sum(Re seq)+Sum(Im seq)*<i>
proof
  assume that
A1: Re seq is summable and
A2: Im seq is summable;
  Partial_Sums(Im seq) is convergent by A2;
  then
A3: Im Partial_Sums(seq) is convergent by Th26;
  Partial_Sums(Re seq) is convergent by A1;
  then
A4: Re Partial_Sums(seq) is convergent by Th26;
  then
A5: lim(Im Partial_Sums(seq))=Im(lim(Partial_Sums(seq))) by A3,Th42;
A6: lim(Re Partial_Sums(seq))=lim(Partial_Sums(Re seq)) & lim(Im
  Partial_Sums( seq))=lim(Partial_Sums(Im seq)) by Th26;
  Partial_Sums(seq) is convergent & lim(Re Partial_Sums(seq))=Re(lim(
  Partial_Sums(seq))) by A4,A3,Th42;
  hence thesis by A6,A5,COMPLEX1:13;
end;
