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;

theorem Th3:
  D c= 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: D c= dom(F.0) and
A2: for x be Element of X st x in D holds F#x is convergent;
A3: for x be Element of X st x in D holds (R_EAL F)#x is convergent
  proof
    let x be Element of X;
    assume x in D;
    then
A4: F#x is convergent by A2;
    (R_EAL F)#x = F#x by MESFUN7C:1;
    hence (R_EAL F)#x is convergent by A4,RINFSUP2:14;
  end;
  for n be Nat holds (R_EAL G).n = ((R_EAL F).n)|D by Def1;
  hence (lim F)|D = lim G by A1,A3,MESFUNC9:19;
end;
