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 Th14:
  F is with_the_same_dom & dom f c= dom(F.0) & x in dom f & f.x =
  Sum(F#x) implies f.x = lim((Partial_Sums F)#x)
proof
  assume that
A1: F is with_the_same_dom & dom f c= dom(F.0) & x in dom f and
A2: f.x = Sum(F#x);
A3: dom Partial_Sums(F#x) = NAT & dom ((Partial_Sums F)#x) = NAT by
FUNCT_2:def 1;
  for n be object st n in NAT
   holds (Partial_Sums(F#x)).n = ((Partial_Sums F)#x).n by A1,Th12;
  hence f.x = lim((Partial_Sums F)#x) by A2,A3,FUNCT_1:2;
end;
