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 Th1:
  x in D & F#x is convergent implies (F||D)#x is convergent
proof
  set G = F||D;
  assume
A1: x in D;
A2: (R_EAL G)#x = G#x by MESFUN7C:1;
  assume F#x is convergent;
  then R_EAL(F#x) is convergent_to_finite_number by RINFSUP2:14;
  then
A3: (R_EAL F)#x is convergent_to_finite_number by MESFUN7C:1;
  for n be Nat holds (R_EAL G).n = ((R_EAL F).n)|D by Def1;
  then (R_EAL G)#x is convergent_to_finite_number by A1,A3,MESFUNC9:12;
  hence thesis by A2,RINFSUP2:15;
end;
