reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S,
  F,G for Functional_Sequence of X,ExtREAL,
  I for ExtREAL_sequence,
  f,g for PartFunc of X,ExtREAL,
  seq, seq1, seq2 for ExtREAL_sequence,
  p for ExtReal,
  n,m for Nat,
  x for Element of X,
  z,D for set;

theorem Th21:
  E c= dom(F.0) & G is with_the_same_dom & (for x be Element of X
  st x in E holds F#x is summable) & (for n be Nat holds G.n= (F.n)|E) implies
  for x be Element of X st x in E holds G#x is summable
proof
  assume that
A1: E c= dom(F.0) and
A2: G is with_the_same_dom and
A3: for x be Element of X st x in E holds F#x is summable and
A4: for n be Nat holds G.n= (F.n)|E;
  let x be Element of X;
  assume
A5: x in E;
  dom((F.0)|E) = E by A1,RELAT_1:62;
  then
A6: E = dom(G.0) by A4;
  for n be Element of NAT holds (F#x).n = (G#x).n
  proof
    let n be Element of NAT;
    dom(G.n) = E by A2,A6;
    then x in dom((F.n)|E) by A4,A5;
    then ((F.n)|E).x = (F.n).x by FUNCT_1:47;
    then
A7: (G.n).x = (F.n).x by A4;
    (F#x).n = (F.n).x by MESFUNC5:def 13;
    hence thesis by A7,MESFUNC5:def 13;
  end;
  then
A8: Partial_Sums(F#x) = Partial_Sums(G#x) by FUNCT_2:63;
  F#x is summable by A3,A5;
  then Partial_Sums(F#x) is convergent;
  hence thesis by A8;
end;
