reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,COMPLEX,
  r for Real,
  k for Real,
  n for Nat,
  E for Element of S;

theorem Th25:
  for f be with_the_same_dom Functional_Sequence of X,COMPLEX st (
for x be Element of X st x in dom(f.0) holds f#x is convergent) holds lim Re f
  = Re lim f & lim Im f = Im lim f
proof
  let f be with_the_same_dom Functional_Sequence of X,COMPLEX;
  dom lim Re f = dom((Re f).0) by MESFUNC8:def 9;
  then
A1: dom lim Re f = dom(f.0) by Def11;
A2: dom Re lim f = dom lim f by COMSEQ_3:def 3;
  then
A3: dom lim Re f = dom Re lim f by A1,Def10;
  assume
A4: for x be Element of X st x in dom(f.0) holds f#x is convergent;
A5: now
    let x be Element of X;
    assume
A6: x in dom lim Re f;
    then
A7: f#x is convergent by A4,A1;
    then Re(f#x) is convergent;
    then
A8: (Re f)#x is convergent by A1,A6,Th23;
    (lim Re f).x = lim R_EAL((Re f)#x) by A6,Th14
      .= lim((Re f)#x) by A8,RINFSUP2:14;
    then (lim Re f).x = lim(Re(f#x)) by A1,A6,Th23;
    then
A9: (lim Re f).x = Re lim(f#x) by A7,COMSEQ_3:41;
    (Re lim f).x = Re((lim f).x) by A3,A6,COMSEQ_3:def 3;
    hence (lim Re f).x = (Re lim f).x by A2,A3,A6,A9,Def10;
  end;
  Re lim f is PartFunc of X,ExtREAL by NUMBERS:31,RELSET_1:7;
  hence lim Re f = Re lim f by A3,A5,PARTFUN1:5;
  dom lim Im f = dom((Im f).0) by MESFUNC8:def 9;
  then
A10: dom lim Im f = dom(f.0) by Def12;
A11: dom Im lim f = dom lim f by COMSEQ_3:def 4;
  then
A12: dom lim Im f = dom Im lim f by A10,Def10;
A13: now
    let x be Element of X;
    assume
A14: x in dom lim Im f;
    then
A15: f#x is convergent by A4,A10;
    then Im(f#x) is convergent;
    then
A16: (Im f)#x is convergent by A10,A14,Th23;
    (lim Im f).x = lim R_EAL((Im f)#x) by A14,Th14
      .= lim((Im f)#x) by A16,RINFSUP2:14;
    then (lim Im f).x = lim Im(f#x) by A10,A14,Th23;
    then
A17: (lim Im f).x = Im lim(f#x) by A15,COMSEQ_3:41;
    (Im lim f).x = Im((lim f).x) by A12,A14,COMSEQ_3:def 4;
    hence (lim Im f).x = (Im lim f).x by A11,A12,A14,A17,Def10;
  end;
  Im lim f is PartFunc of X,ExtREAL by NUMBERS:31,RELSET_1:7;
  hence thesis by A12,A13,PARTFUN1:5;
end;
