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 Th35:
  F is with_the_same_dom & D c= dom(F.0) & x in D implies (
  Partial_Sums(F#x) is convergent iff (Partial_Sums F)#x is convergent )
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((Re F).0) by A2,MESFUN7C:def 11;
A5: D c= dom((Im F).0) by A2,MESFUN7C:def 12;
A6: dom((Partial_Sums F).0) = dom(F.0) & Partial_Sums F is with_the_same_dom
  by A1,Th32,Th34;
A7: Re F is with_the_same_dom by A1;
  then
A8: Im F is with_the_same_dom by Th25;
  hereby
    assume
A9: Partial_Sums(F#x) is convergent;
    then Im(Partial_Sums(F#x)) is convergent;
    then Partial_Sums Im(F#x) is convergent by COMSEQ_3:26;
    then Partial_Sums((Im F)#x) is convergent by A1,A2,A3,MESFUN7C:23;
    then (Partial_Sums Im F)#x is convergent by A3,A8,A5,Th13;
    then (Im(Partial_Sums F))#x is convergent by Th29;
    then
A10: Im((Partial_Sums F)#x) is convergent by A2,A3,A6,MESFUN7C:23;
    Re(Partial_Sums(F#x)) is convergent by A9;
    then Partial_Sums Re(F#x) is convergent by COMSEQ_3:26;
    then Partial_Sums((Re F)#x) is convergent by A1,A2,A3,MESFUN7C:23;
    then (Partial_Sums Re F)#x is convergent by A3,A7,A4,Th13;
    then (Re(Partial_Sums F))#x is convergent by Th29;
    then Re((Partial_Sums F)#x) is convergent by A2,A3,A6,MESFUN7C:23;
    hence (Partial_Sums F)#x is convergent by A10,COMSEQ_3:42;
  end;
  assume
A11: (Partial_Sums F)#x is convergent;
  then Im((Partial_Sums F)#x) is convergent;
  then
A12: (Im(Partial_Sums F))#x is convergent by A2,A3,A6,MESFUN7C:23;
A13: (Im F)#x = Im(F#x) by A1,A2,A3,MESFUN7C:23;
  Re((Partial_Sums F)#x) is convergent by A11;
  then
A14: (Re(Partial_Sums F))#x is convergent by A2,A3,A6,MESFUN7C:23;
  Partial_Sums((Im F)#x) is convergent iff (Partial_Sums Im F)#x is
  convergent by A3,A8,A5,Th13;
  then
A15: Im(Partial_Sums(F#x)) is convergent by A12,A13,Th29,COMSEQ_3:26;
A16: (Re F)#x = Re(F#x) by A1,A2,A3,MESFUN7C:23;
  Partial_Sums((Re F)#x) is convergent iff (Partial_Sums Re F)#x is
  convergent by A3,A7,A4,Th13;
  then Re(Partial_Sums(F#x)) is convergent by A14,A16,Th29,COMSEQ_3:26;
  hence Partial_Sums(F#x) is convergent by A15,COMSEQ_3:42;
end;
