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 Th26:
  Partial_Sums Re seq = Re (Partial_Sums seq) & Partial_Sums Im
  seq = Im (Partial_Sums seq)
proof
  defpred P[Nat] means
Partial_Sums(Re seq).$1 = Re Partial_Sums(
  seq).$1;
  defpred R[Nat] means
Partial_Sums(Im seq).$1 = Im Partial_Sums(seq).$1;
A1: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume P[k];
    then Partial_Sums(Re seq).(k+1) =Re Partial_Sums(seq).k + Re seq.(k+1) by
SERIES_1:def 1
      .=Re Partial_Sums(seq).k +Re(seq.(k+1)) by Def5
      .=Re(Partial_Sums(seq).k) +Re(seq.(k+1)) by Def5
      .=Re(Partial_Sums(seq).k + seq.(k+1)) by COMPLEX1:8
      .=Re(Partial_Sums(seq).(k+1)) by SERIES_1:def 1
      .=Re Partial_Sums(seq).(k+1) by Def5;
    hence thesis;
  end;
A2: for k be Nat st R[k] holds R[k+1]
  proof
    let k be Nat;
    assume R[k];
    then
    Partial_Sums(Im seq).(k+1) = Im Partial_Sums(seq).k + Im seq.(k+1) by
SERIES_1:def 1
      .=Im Partial_Sums(seq).k +Im(seq.(k+1)) by Def6
      .=Im(Partial_Sums(seq).k) +Im(seq.(k+1)) by Def6
      .=Im(Partial_Sums(seq).k + seq.(k+1)) by COMPLEX1:8
      .=Im(Partial_Sums(seq).(k+1)) by SERIES_1:def 1
      .=Im Partial_Sums(seq).(k+1) by Def6;
    hence thesis;
  end;
  Partial_Sums(Im seq).0 = Im seq.0 by SERIES_1:def 1
    .= Im(seq.0) by Def6
    .= Im(Partial_Sums(seq).0) by SERIES_1:def 1
    .= Im Partial_Sums(seq).0 by Def6;
  then
A3: R[0];
A4: for n being Nat holds R[n] from NAT_1:sch 2(A3,A2);
  Partial_Sums(Re seq).0 = Re seq.0 by SERIES_1:def 1
    .= Re(seq.0) by Def5
    .= Re(Partial_Sums(seq).0) by SERIES_1:def 1
    .= Re Partial_Sums(seq).0 by Def5;
  then
A5: P[0];
  for n being Nat holds P[n] from NAT_1:sch 2(A5,A1);
  hence thesis by A4;
end;
