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 & dom f c= dom(F.0) & x in dom f & F#x is
  summable & f.x = Sum(F#x) implies f.x = lim((Partial_Sums F)#x)
proof
  assume that
A1: F is with_the_same_dom and
A2: dom f c= dom(F.0) and
A3: x in dom f and
A4: F#x is summable and
A5: f.x = Sum(F#x);
  dom Re f = dom f by COMSEQ_3:def 3;
  then
A6: dom Re f c= dom((Re F).0) by A2,MESFUN7C:def 11;
  Partial_Sums(F#x) is convergent by A4;
  then (Partial_Sums F)#x is convergent by A1,A2,A3,Th35;
  then
A7: lim(Re((Partial_Sums F)#x)) = Re(lim((Partial_Sums F)#x)) & lim(Im((
  Partial_Sums F)#x)) = Im(lim((Partial_Sums F)#x)) by COMSEQ_3:41;
  dom Im f = dom f by COMSEQ_3:def 4;
  then
A8: dom Im f c= dom((Im F).0) by A2,MESFUN7C:def 12;
A9: x in dom Im f by A3,COMSEQ_3:def 4;
  then
A10: (Im f).x = Im(f.x) by COMSEQ_3:def 4;
A11: Partial_Sums F is with_the_same_dom & dom((Partial_Sums F).0) = dom(F.0
  ) by A1,Th32,Th34;
  then (Re(Partial_Sums F))#x = Re((Partial_Sums F)#x) by A2,A3,MESFUN7C:23;
  then
A12: lim((Partial_Sums Re F)#x) = lim(Re((Partial_Sums F)#x)) by Th29;
  (Im(Partial_Sums F))#x = Im((Partial_Sums F)#x) by A2,A3,A11,MESFUN7C:23;
  then
A13: lim((Partial_Sums Im F)#x) = lim(Im((Partial_Sums F)#x)) by Th29;
A14: x in dom Re f by A3,COMSEQ_3:def 3;
  then
A15: (Re f).x = Re(f.x) by COMSEQ_3:def 3;
A16: Re F is with_the_same_dom by A1;
  then
A17: Im F is with_the_same_dom by Th25;
  Re(F#x) = (Re F)#x & Im(F#x) = (Im F)#x by A1,A2,A3,MESFUN7C:23;
  then
A18: Sum(F#x) = Sum((Re F)#x) + Sum((Im F)#x)*<i> by A4,COMSEQ_3:53;
  then Re(Sum(F#x)) = Sum((Re F)#x) by COMPLEX1:12;
  then (Re f).x = Sum((Re F)#x) by A5,A14,COMSEQ_3:def 3;
  then
A19: (Re f).x = lim((Partial_Sums Re F)#x) by A16,A6,A14,Th14;
  Im(Sum(F#x)) = Sum((Im F)#x) by A18,COMPLEX1:12;
  then (Im f).x = Sum((Im F)#x) by A5,A9,COMSEQ_3:def 4;
  then
A20: (Im f).x = lim((Partial_Sums Im F)#x) by A17,A8,A9,Th14;
  thus f.x = Re(f.x) + Im(f.x)*<i> by COMPLEX1:13
    .= lim((Partial_Sums F)#x) by A15,A10,A19,A20,A7,A12,A13,COMPLEX1:13;
end;
