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 Th53:
  seq is summable implies Re seq is summable & Im seq is summable
  & Sum(seq)= Sum(Re seq)+Sum(Im seq)*<i>
proof
A1: lim(Re Partial_Sums(seq))=lim(Partial_Sums(Re seq)) & lim(Im
  Partial_Sums( seq))=lim(Partial_Sums(Im seq)) by Th26;
  assume
A2: seq is summable;
  then Re Partial_Sums(seq) is convergent;
  then
A3: Partial_Sums(Re seq) is convergent by Th26;
  Im Partial_Sums(seq) is convergent by A2;
  then
A4: Partial_Sums(Im seq) is convergent by Th26;
  lim(Re Partial_Sums(seq))=Re(lim(Partial_Sums(seq))) & lim(Im
  Partial_Sums( seq))=Im(lim(Partial_Sums(seq))) by A2,Th41;
  hence thesis by A3,A4,A1,COMPLEX1:13;
end;
