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 Th55:
  seq1 is summable & seq2 is summable implies seq1-seq2 is
  summable & Sum(seq1-seq2)= Sum(seq1)-Sum(seq2)
proof
  assume
A1: seq1 is summable & seq2 is summable;
  then
A2: Partial_Sums(seq1)-Partial_Sums(seq2) is convergent;
A3: Partial_Sums(seq1)-Partial_Sums(seq2) =Partial_Sums(seq1-seq2) by Th28;
  Sum(seq1-seq2)=lim(Partial_Sums(seq1)-Partial_Sums(seq2)) by Th28
    .=lim(Partial_Sums(seq1))-lim(Partial_Sums(seq2)) by A1,COMSEQ_2:26;
  hence thesis by A2,A3;
end;
