reserve q,th,r for Real,
  a,b,p for Real,
  w,z for Complex,
  k,l,m,n,n1,n2 for Nat,
  seq,seq1,seq2,cq1 for Complex_Sequence,
  rseq,rseq1,rseq2 for Real_Sequence,
  rr for set,
  hy1 for 0-convergent non-zero Real_Sequence;
reserve d for Real;
reserve th,th1,th2 for Real;

theorem Th34:
  for th holds
  Partial_Sums(th P_sin).n = Partial_Sums(Im ((th*<i>) ExpSeq)).(2*n+1) &
  Partial_Sums(th P_cos).n = Partial_Sums(Re ((th*<i>) ExpSeq)).(2*n)
proof
  let th;
 now
A1: Partial_Sums(th P_sin).0= (th P_sin).0 by SERIES_1:def 1
      .= (-1)|^ 0 * (th)|^ (2*0+1)/((2*0+1)!) by Def20;
A2: ((2*0+1)!)= 0! * 1 by NEWTON:15
      .= 1 by NEWTON:12;
A3: (-1)|^ 0 = (-1) GeoSeq.0 by PREPOWER:def 1
      .= 1 by PREPOWER:3;
A4: Partial_Sums(th P_cos).0= (th P_cos).0 by SERIES_1:def 1
      .=(-1)|^ 0 * (th)|^ (2*0)/((2*0)!) by Def21
      .= 1* th GeoSeq.0 / 1 by A3,NEWTON:12,PREPOWER:def 1
      .=1 by PREPOWER:3;
    A5: (
Im ((th*<i>) ExpSeq)).0= Im (((th*<i>) ExpSeq).0) & (Im ((th*<i>) ExpSeq)).
    1= Im (((th*<i>) ExpSeq).1) by COMSEQ_3:def 6;
A6: (th*<i>) = 0+th*<i>;
A7: Partial_Sums(Im ((th*<i>) ExpSeq)).(2*0+1)
    = Partial_Sums(Im ((th*<i>) ExpSeq)).0 +
    (Im ((th*<i>) ExpSeq)).1 by SERIES_1:def 1
      .= (Im ((th*<i>) ExpSeq)).0+
    (Im ((th*<i>) ExpSeq)).1 by SERIES_1:def 1
      .= 0 + Im (((th*<i>) ExpSeq).(0+1)) by A5,Th3,COMPLEX1:6
      .=0 + Im ((((th*<i>) ExpSeq).0)* (th*<i>) /(0+1+0*<i>)) by Th3
      .= Im (1* (th*<i>) /1) by Th3
      .=th by A6,COMPLEX1:12;
    defpred X[Nat] means
    Partial_Sums(th P_sin).$1 = Partial_Sums(Im ((th*<i>) ExpSeq)).(2*$1+1) &
    Partial_Sums(th P_cos).$1 = Partial_Sums(Re ((th*<i>) ExpSeq)).(2*$1);
 Partial_Sums(Re ((th*<i>) ExpSeq)).(2*0)
    = (Re ((th*<i>) ExpSeq)).0 by SERIES_1:def 1
      .= Re (((th*<i>) ExpSeq).0) by COMSEQ_3:def 5
      .=1 by Th3,COMPLEX1:6;
then A8: X[0] by A1,A2,A3,A4,A7;
A9: for k st X[k] holds X[k+1]
    proof
      let k be Nat;
      assume that
A10:  Partial_Sums(th P_sin).k = Partial_Sums(Im ((th*<i>) ExpSeq)).(2*k+1) and
A11:  Partial_Sums(th P_cos).k = Partial_Sums(Re ((th*<i>) ExpSeq)).(2*k);
  Partial_Sums(Im ((th*<i>) ExpSeq)).(2*(k+1)+1)
      = Partial_Sums(Im ((th*<i>) ExpSeq)).((2*k+1)+1) +
      (Im ((th*<i>) ExpSeq)).(2*(k+1)+1) by SERIES_1:def 1
        .= Partial_Sums(th P_sin).k +
      (Im((th*<i>) ExpSeq)).(2*(k+1))+(Im ((th*<i>) ExpSeq)).(2*(k+1)+1)
      by A10,SERIES_1:def 1
        .= Partial_Sums(th P_sin).k +
      Im (((th*<i>) ExpSeq).(2*(k+1)))+(Im((th*<i>) ExpSeq)).(2*(k+1)+1)
      by COMSEQ_3:def 6
        .=Partial_Sums(th P_sin).k +
      Im (((th*<i>) ExpSeq).(2*(k+1)))+Im(((th*<i>) ExpSeq).(2*(k+1)+1))
      by COMSEQ_3:def 6
        .= Partial_Sums(th P_sin).k +
      Im((th*<i>) |^ (2*(k+1))/((2*(k+1))!))+
      Im(((th*<i>) ExpSeq).(2*(k+1)+1)) by Def4
        .= Partial_Sums(th P_sin).k +
      Im((th*<i>) |^ (2*(k+1))/((2*(k+1))!))+
      Im((th*<i>) |^ (2*(k+1)+1)/((2*(k+1)+1)!)) by Def4
        .= Partial_Sums(th P_sin).k +
      Im( (-1)|^ (k+1)* (th |^ (2*(k+1)))/ ((2*(k+1))!))
      +Im((th*<i>) |^ (2*(k+1)+1)/((2*(k+1)+1)!)) by Th33
        .= Partial_Sums(th P_sin).k +
      Im( (-1)|^ (k+1)* (th |^ (2*(k+1)))/ ((2*(k+1))!))
      + Im( (-1)|^ (k+1)* (th |^ (2*(k+1)+1))*<i>/((2*(k+1)+1)!)) by Th33;
then   Partial_Sums(Im ((th*<i>) ExpSeq)).(2*(k+1)+1)
      = Partial_Sums(th P_sin).k +
      Im( (((-1)|^ (k+1)* (th |^ (2*(k+1))))/((2*(k+1))!)+0/((2*(k+1))!)*<i>))
      + Im( ((-1)|^ (k+1)* (th |^ (2*(k+1)+1))*<i>)/((2*(k+1)+1)!));
then   Partial_Sums(Im ((th*<i>) ExpSeq)).(2*(k+1)+1)
      = Partial_Sums(th P_sin).k + 0 + Im(0/((2*(k+1)+1)!)+
      ((-1)|^ (k+1)* (th |^ (2*(k+1)+1)))/((2*(k+1)+1)!)*<i>)
      by COMPLEX1:12
        .= Partial_Sums(th P_sin).k + (0/((2*(k+1))!)) +
      (-1)|^ (k+1)* (th |^ (2*(k+1)+1))/((2*(k+1)+1)!) by COMPLEX1:12;
then A12:  Partial_Sums(Im ((th*<i>) ExpSeq)).(2*(k+1)+1)
      = Partial_Sums(th P_sin).k + th P_sin.(k+1) by Def20
        .= Partial_Sums(th P_sin).(k+1) by SERIES_1:def 1;
  Partial_Sums(Re ((th*<i>) ExpSeq)).(2*(k+1))
      = Partial_Sums(Re ((th*<i>) ExpSeq)).(2*k+1)+
      (Re ((th*<i>) ExpSeq)).((2*k+1)+1) by SERIES_1:def 1
        .=Partial_Sums(th P_cos).k + (Re ((th*<i>) ExpSeq)).(2*k+1)+
      (Re ((th*<i>) ExpSeq)).((2*k+1)+1) by A11,SERIES_1:def 1
        .= Partial_Sums(th P_cos).k + Re (((th*<i>) ExpSeq).(2*k+1))+
      (Re ((th*<i>) ExpSeq)).((2*k+1)+1) by COMSEQ_3:def 5
        .= Partial_Sums(th P_cos).k + Re (((th*<i>) ExpSeq).(2*k+1))+
      Re (((th*<i>) ExpSeq).((2*k+1)+1)) by COMSEQ_3:def 5
        .=Partial_Sums(th P_cos).k + Re ((th*<i>)|^ (2*k+1)/ ((2*k+1)!))+
      Re (((th*<i>) ExpSeq).((2*k+1)+1)) by Def4
        .= Partial_Sums(th P_cos).k + Re ((th*<i>)|^ (2*k+1)/ ((2*k+1)!))+
      Re ((th*<i>)|^ (2*(k+1))/(((2*k+1)+1)!)) by Def4
        .= Partial_Sums(th P_cos).k +
      Re ( (-1)|^ k* (th |^ (2*k+1))*<i>/((2*k+1)!))+
      Re ((th*<i>)|^ (2*(k+1))/((2*(k+1))!)) by Th33
        .= Partial_Sums(th P_cos).k +
      Re ( (-1)|^ k* (th |^ (2*k+1))*<i>/((2*k+1)!))+
      Re ((-1)|^ (k+1)* (th |^ (2*(k+1)))/((2*(k+1))!)) by Th33;
then
  Partial_Sums(Re ((th*<i>) ExpSeq)).(2*(k+1)) = Partial_Sums(th P_cos).k +
      Re ( (0/((2*k+1)!)+((-1)|^ k* (th |^ (2*k+1)))/((2*k+1)!)*<i>)) +
      Re ((-1)|^ (k+1)* (th |^ (2*(k+1)))/((2*(k+1))!));
then   Partial_Sums(Re ((th*<i>) ExpSeq)).(2*(k+1))
      = Partial_Sums(th P_cos).k +0/((2*k+1)!)+
      Re ((-1)|^ (k+1)* (th |^ (2*(k+1)))/((2*(k+1))!)+(0/((2*(k+1))!))*<i>)
      by COMPLEX1:12
        .= Partial_Sums(th P_cos).k +0/((2*k+1)!)+
      (-1)|^ (k+1)* (th |^ (2*(k+1)))/((2*(k+1))!) by COMPLEX1:12;
then   Partial_Sums(Re ((th*<i>) ExpSeq)).(2*(k+1))
      = Partial_Sums(th P_cos).k + th P_cos.(k+1) by Def21
        .= Partial_Sums(th P_cos).(k+1) by SERIES_1:def 1;
      hence thesis by A12;
    end;
    thus for n holds X[n] from NAT_1:sch 2(A8,A9);
  end;
  hence thesis;
end;
