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 Th13:
  F is with_the_same_dom & D c= dom(F.0) & x in D implies (
  Partial_Sums(F#x) is convergent iff (Partial_Sums F)#x is convergent )
proof
  assume
A1: F is with_the_same_dom & D c= dom(F.0) & x in D;
A2: R_EAL(F#x) = (R_EAL F)#x by MESFUN7C:1;
  Partial_Sums R_EAL F = R_EAL(Partial_Sums F) by Th7;
  then
A3: (Partial_Sums(R_EAL F))#x= (Partial_Sums F)#x by MESFUN7C:1;
A4: Partial_Sums(F#x) = R_EAL(Partial_Sums(F#x))
    .= Partial_Sums(R_EAL(F#x)) by Th5;
A5: R_EAL F is additive by Th9;
  hereby
    assume Partial_Sums(F#x) is convergent;
    then Partial_Sums(R_EAL(F#x)) is convergent_to_finite_number by A4,
RINFSUP2:14;
    then
    (Partial_Sums(R_EAL F))#x is convergent_to_finite_number by A1,A5,A2,
MESFUNC9:33;
    hence (Partial_Sums F)#x is convergent by A3,RINFSUP2:15;
  end;
  assume (Partial_Sums F)#x is convergent;
  then (Partial_Sums(R_EAL F))#x is convergent_to_finite_number by A3,
RINFSUP2:14;
  then Partial_Sums((R_EAL F)#x) is convergent_to_finite_number by A1,A5,
MESFUNC9:33;
  hence Partial_Sums(F#x) is convergent by A4,A2,RINFSUP2:15;
end;
