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;

theorem Th6:
  (for x be Element of X st x in E holds F#x is summable) implies
  for x be Element of X st x in E holds (F||E)#x is summable
proof
  set G = F||E;
  assume
A1: for x be Element of X st x in E holds F#x is summable;
  let x be Element of X;
  assume
A2: x in E;
  for n be Element of NAT holds (F#x).n = (G#x).n
  proof
    let n be Element of NAT;
    (F#x).n = (F.n).x by SEQFUNC:def 10;
    then (F#x).n = ((F.n)|E).x by A2,FUNCT_1:49;
    then (F#x).n = (G.n).x by Def1;
    hence (F#x).n = (G#x).n by SEQFUNC:def 10;
  end;
  then
A3: Partial_Sums(F#x) = Partial_Sums(G#x) by FUNCT_2:63;
  F#x is summable by A1,A2;
  then Partial_Sums(F#x) is convergent;
  hence G#x is summable by A3;
end;
