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 Th28:
  Partial_Sums(seq1)-Partial_Sums(seq2) = Partial_Sums(seq1-seq2)
proof
A1: Im(Partial_Sums(seq1)-Partial_Sums(seq2)) =Im Partial_Sums(seq1)-Im
  Partial_Sums(seq2) by Th18
    .=Partial_Sums Im seq1-Im Partial_Sums(seq2) by Th26
    .=Partial_Sums Im seq1-Partial_Sums Im seq2 by Th26
    .=Partial_Sums(Im seq1-Im seq2) by SERIES_1:6
    .=Partial_Sums(Im (seq1-seq2)) by Th18
    .=Im Partial_Sums(seq1-seq2) by Th26;
  Re (Partial_Sums(seq1)-Partial_Sums(seq2)) =Re Partial_Sums(seq1)-Re
  Partial_Sums(seq2) by Th18
    .=Partial_Sums Re seq1-Re Partial_Sums(seq2) by Th26
    .=Partial_Sums Re seq1-Partial_Sums Re seq2 by Th26
    .=Partial_Sums(Re seq1-Re seq2) by SERIES_1:6
    .=Partial_Sums(Re (seq1-seq2)) by Th18
    .=Re Partial_Sums(seq1-seq2) by Th26;
  hence thesis by A1,Th14;
end;
