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 Th54:
  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 Th27;
  Sum(seq1+seq2)=lim(Partial_Sums(seq1)+Partial_Sums(seq2)) by Th27
    .=lim(Partial_Sums(seq1))+lim(Partial_Sums(seq2)) by A1,COMSEQ_2:14;
  hence thesis by A2,A3;
end;
