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 = dom(F.0) & (for x be Element of X st x in
  D holds F#x is convergent) implies (lim F)|D = lim (F||D)
proof
  set G = F||D;
  assume that
A1: F is with_the_same_dom and
A2: D = dom(F.0) and
A3: for x be Element of X st x in D holds F#x is convergent;
A4: Re G = (Re F)||D by Th21;
A5: now
    let x be Element of X;
    dom((F.0)|D) = D by A2,RELAT_1:62;
    then
A6: dom(G.0) = D by Def1;
    assume
A7: x in dom(G.0);
    then F#x is convergent by A3,A6;
    hence G#x is convergent by A1,A2,A7,A6,Th23;
  end;
A8: for x be Element of X st x in D holds (Re F)#x is convergent
  proof
    let x be Element of X;
    assume
A9: x in D;
    then F#x is convergent Complex_Sequence by A3;
    then Re(F#x) is convergent;
    hence (Re F)#x is convergent by A1,A2,A9,MESFUN7C:23;
  end;
  D c= dom((Re F).0) by A2,MESFUN7C:def 11;
  then (lim Re F)|D = lim Re G by A4,A8,Th3;
  then
A10: (Re lim F)|D = lim Re G by A1,A2,A3,MESFUN7C:25;
A11: G is with_the_same_dom by A1,Th2;
  then lim Re G = Re(lim G) by A5,MESFUN7C:25;
  then
A12: Re((lim F)|D) = Re(lim G) by A10,Th20;
A13: for x be Element of X st x in D holds (Im F)#x is convergent
  proof
    let x be Element of X;
    assume
A14: x in D;
    then F#x is convergent Complex_Sequence by A3;
    then Im(F#x) is convergent;
    hence (Im F)#x is convergent by A1,A2,A14,MESFUN7C:23;
  end;
A15: Im G = (Im F)||D by Th22;
  D c= dom((Im F).0) by A2,MESFUN7C:def 12;
  then (lim Im F)|D = lim Im G by A15,A13,Th3;
  then
A16: (Im lim F)|D = lim Im G by A1,A2,A3,MESFUN7C:25;
  lim Im G = Im(lim G) by A11,A5,MESFUN7C:25;
  then
A17: Im((lim F)|D) = Im(lim G) by A16,Th20;
  thus lim G = Re(lim G) + <i>(#)(Im(lim G)) by MESFUN6C:8
    .= (lim F)|D by A12,A17,MESFUN6C:8;
end;
