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 Th23:
  F is with_the_same_dom & D c= dom(F.0) & x in D implies (F#x is
  convergent implies (F||D)#x is convergent)
proof
  set G = F||D;
  assume that
A1: F is with_the_same_dom and
A2: D c= dom(F.0) and
A3: x in D;
  Re G = (Re F)||D by Th21;
  then
A4: (Re F)#x is convergent implies (Re G)#x is convergent by A3,Th1;
  dom((Re F).0) = dom(F.0) by MESFUN7C:def 11;
  then dom(((Re F).0)|D) = D by A2,RELAT_1:62;
  then dom((Re G).0) = D by Lm1;
  then
A5: dom(G.0) = D by MESFUN7C:def 11;
  G is with_the_same_dom by A1,Th2;
  then
A6: (Re G)#x = Re(G#x) & (Im G)#x = Im(G#x) by A3,A5,MESFUN7C:23;
  Im G = (Im F)||D by Th22;
  then
A7: (Im F)#x is convergent implies (Im G)#x is convergent by A3,Th1;
  (Re F)#x = Re(F#x) & (Im F)#x = Im(F#x) by A1,A2,A3,MESFUN7C:23;
  hence thesis by A4,A7,A6,COMSEQ_3:42;
end;
