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
  dom((Partial_Sums F).n) = meet{dom(F.k) where k is Element of NAT : k <= n}
proof
  now
    let A be object;
    assume A in {dom((Re F).k) where k is Element of NAT : k <= n};
    then consider i be Element of NAT such that
A1: A = dom((Re F).i) and
A2: i <= n;
    A = dom(F.i) by A1,MESFUN7C:def 11;
    hence A in {dom(F.k) where k is Element of NAT : k <= n} by A2;
  end;
  then
A3: {dom((Re F).k) where k is Element of NAT : k <= n} c= {dom(F.k) where k
  is Element of NAT : k <= n} by TARSKI:def 3;
  now
    let A be object;
    assume A in {dom(F.k) where k is Element of NAT : k <= n};
    then consider i be Element of NAT such that
A4: A = dom(F.i) and
A5: i <= n;
    A = dom((Re F).i) by A4,MESFUN7C:def 11;
    hence A in {dom((Re F).k) where k is Element of NAT : k <= n} by A5;
  end;
  then
A6: {dom(F.k) where k is Element of NAT : k <= n} c= {dom((Re F).k) where k
  is Element of NAT : k <= n} by TARSKI:def 3;
  dom((Partial_Sums Re F).n) = dom((Re(Partial_Sums F)).n) by Th29;
  then
A7: dom((Partial_Sums Re F).n) = dom((Partial_Sums F).n) by MESFUN7C:def 11;
  dom((Partial_Sums Re F).n) = meet{dom((Re F).k) where k is Element of NAT
  : k <= n} by Th10;
  hence thesis by A7,A3,A6,XBOOLE_0:def 10;
end;
