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
  F is with_the_same_dom & D c= dom(F.0) & x in D implies (Partial_Sums(
  F#x)).n = ((Partial_Sums F)#x).n
proof
  assume that
A1: F is with_the_same_dom and
A2: D c= dom(F.0) and
A3: x in D;
A4: D c= dom((Im F).0) by A2,MESFUN7C:def 12;
  dom((Partial_Sums F).n) = dom(F.0) by A1,Th32;
  then
A5: x in dom((Partial_Sums F).n) by A2,A3;
  then
A6: x in dom Re((Partial_Sums F).n) by COMSEQ_3:def 3;
A7: Re F is with_the_same_dom by A1;
  then Im F is with_the_same_dom by Th25;
  then
A8: (Partial_Sums((Im F)#x)).n = ((Partial_Sums Im F)#x).n by A3,A4,Th12;
  D c= dom((Re F).0) by A2,MESFUN7C:def 11;
  then
A9: (Partial_Sums((Re F)#x)).n = ((Partial_Sums Re F)#x).n by A3,A7,Th12;
A10: Re((Partial_Sums(F#x)).n) = (Re(Partial_Sums(F#x))).n by COMSEQ_3:def 5
    .= (Partial_Sums Re(F#x)).n by COMSEQ_3:26
    .= (Partial_Sums((Re F)#x)).n by A1,A2,A3,MESFUN7C:23
    .= ((Partial_Sums Re F).n).x by A9,SEQFUNC:def 10
    .= ((Re(Partial_Sums F)).n).x by Th29
    .= (Re((Partial_Sums F).n)).x by MESFUN7C:24
    .= Re(((Partial_Sums F).n).x) by A6,COMSEQ_3:def 3
    .= Re(((Partial_Sums F)#x).n) by MESFUN7C:def 9;
A11: x in dom Im((Partial_Sums F).n) by A5,COMSEQ_3:def 4;
A12: Im((Partial_Sums(F#x)).n) = (Im(Partial_Sums(F#x))).n by
COMSEQ_3:def 6
    .= (Partial_Sums Im(F#x)).n by COMSEQ_3:26
    .= (Partial_Sums((Im F)#x)).n by A1,A2,A3,MESFUN7C:23
    .= ((Partial_Sums Im F).n).x by A8,SEQFUNC:def 10
    .= ((Im(Partial_Sums F)).n).x by Th29
    .= (Im((Partial_Sums F).n)).x by MESFUN7C:24
    .= Im(((Partial_Sums F).n).x) by A11,COMSEQ_3:def 4
    .= Im(((Partial_Sums F)#x).n) by MESFUN7C:def 9;
  thus (Partial_Sums(F#x)).n = Re((Partial_Sums(F#x)).n) + ( Im((Partial_Sums(
  F#x)).n) )*<i> by COMPLEX1:13
    .= ((Partial_Sums F)#x).n by A10,A12,COMPLEX1:13;
end;
