reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S,
  F for Functional_Sequence of X,REAL,

  f for PartFunc of X,REAL,
  seq for Real_Sequence,
  n,m for Nat,
  x for Element of X,
  z,D for set;
reserve i for Element of NAT;
reserve F for Functional_Sequence of X,COMPLEX,
  f for PartFunc of X,COMPLEX,
  A for set;

theorem Th29:
  Partial_Sums Re F = Re Partial_Sums F & Partial_Sums Im F = Im Partial_Sums F
proof
  defpred P[Nat] means
   Partial_Sums(Re F).$1 = (Re Partial_Sums F).$1;
  defpred R[Nat] means
Partial_Sums(Im F).$1 = (Im Partial_Sums F).$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 F).(k+1) =(Re Partial_Sums F).k + Re F.(k+1) by Def2
      .=(Re Partial_Sums F).k +Re(F.(k+1)) by MESFUN7C:24
      .=Re((Partial_Sums F).k) +Re(F.(k+1)) by MESFUN7C:24
      .=Re((Partial_Sums F).k + F.(k+1)) by MESFUN6C:5
      .=Re((Partial_Sums F).(k+1)) by Def3;
    hence thesis by MESFUN7C:24;
  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 F).(k+1) = (Im Partial_Sums F).k + Im F.(k+1) by Def2
      .=(Im Partial_Sums F).k +Im(F.(k+1)) by MESFUN7C:24
      .=Im((Partial_Sums F).k) +Im(F.(k+1)) by MESFUN7C:24
      .=Im((Partial_Sums F).k + F.(k+1)) by MESFUN6C:5
      .=Im((Partial_Sums F).(k+1)) by Def3;
    hence thesis by MESFUN7C:24;
  end;
  Partial_Sums(Im F).0 = Im F.0 by Def2
    .= Im(F.0) by MESFUN7C:24
    .= Im((Partial_Sums F).0) by Def3;
  then
A3: R[ 0 ] by MESFUN7C:24;
A4: for i being Nat holds R[i] from NAT_1:sch 2(A3,A2);
  Partial_Sums(Re F).0 = Re F.0 by Def2
    .= Re(F.0) by MESFUN7C:24
    .= Re(Partial_Sums F.0) by Def3;
  then
A5: P[ 0 ] by MESFUN7C:24;
  for i being Nat holds P[i] from NAT_1:sch 2(A5,A1);
  hence thesis by A4;
end;
