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
  (for n holds seq1.n=seq.0) implies Partial_Sums(seq^\1) = (
  Partial_Sums(seq)^\1) - seq1
proof
  assume
A1: for n holds seq1.n=seq.0;
A2: now
    let n;
    thus Re seq1.n=Re(seq1.n) by Def5
      .=Re(seq.0) by A1
      .=Re seq.0 by Def5;
    thus Im seq1.n=Im(seq1.n) by Def6
      .=Im(seq.0) by A1
      .=Im seq.0 by Def6;
  end;
A3: Im Partial_Sums(seq^\1)=Partial_Sums(Im (seq^\1)) by Th26
    .=Partial_Sums(Im seq^\1) by Th23
    .= (Partial_Sums(Im seq)^\1) - Im seq1 by A2,SERIES_1:11
    .=Im (Partial_Sums(seq)^\1)-Im seq1 by Th32
    .=Im (Partial_Sums(seq)^\1-seq1) by Th18;
  Re Partial_Sums(seq^\1)=Partial_Sums(Re (seq^\1)) by Th26
    .=Partial_Sums(Re seq^\1) by Th23
    .= (Partial_Sums(Re seq)^\1) - Re seq1 by A2,SERIES_1:11
    .=Re (Partial_Sums(seq)^\1)-Re seq1 by Th32
    .=Re (Partial_Sums(seq)^\1-seq1) by Th18;
  hence thesis by A3,Th14;
end;
