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;
reserve F for Functional_Sequence of X,COMPLEX,
  f for PartFunc of X,COMPLEX,
  A for set;

theorem
  z in dom((Partial_Sums F).n) & m <= n implies z in dom((Partial_Sums F
  ).m) & z in dom(F.m)
proof
  assume
A1: z in dom((Partial_Sums F).n) & m <= n;
A2: dom((Partial_Sums F).n) = dom((Re(Partial_Sums F)).n) by MESFUN7C:def 11
    .= dom((Partial_Sums Re F).n) by Th29;
  dom((Partial_Sums Re F).m) = dom((Re(Partial_Sums F)).m) by Th29
    .= dom((Partial_Sums F).m) by MESFUN7C:def 11;
  hence z in dom((Partial_Sums F).m) by A1,A2,Th8;
  z in dom((Re F).m) by A1,A2,Th8;
  hence z in dom(F.m) by MESFUN7C:def 11;
end;
