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 Th29:
  F is additive & F is with_the_same_dom implies dom((Partial_Sums
  F).n) = dom(F.0)
proof
  assume that
A1: F is additive and
A2: F is with_the_same_dom;
  now
    let D be object;
A3: dom(F.0) in {dom(F.k) where k is Element of NAT : k <= n};
    assume D in meet {dom(F.k) where k is Element of NAT : k <= n};
    then D in dom(F.0) by A3,SETFAM_1:def 1;
    hence D in meet {dom(F.0)} by SETFAM_1:10;
  end;
  then
A4: meet{dom(F.k) where k is Element of NAT : k <= n} c= meet{dom(F.0)};
  now
    let D be object;
    assume D in meet {dom(F.0)};
    then
A5: D in dom(F.0) by SETFAM_1:10;
A6: for E be set holds E in {dom(F.k) where k is Element of NAT : k <= n}
    implies D in E
    proof
      let E be set;
      assume E in {dom(F.k) where k is Element of NAT : k <= n};
      then ex k1 be Element of NAT st E = dom(F.k1) & k1 <= n;
      hence thesis by A2,A5;
    end;
    dom(F.0) in {dom(F.k) where k is Element of NAT : k <= n};
    hence D in meet {dom(F.k) where k is Element of NAT : k <= n} by A6,
SETFAM_1:def 1;
  end;
  then
  meet{dom(F.0)} c= meet {dom(F.k) where k is Element of NAT : k <= n};
  then meet{dom(F.k) where k is Element of NAT : k <= n} = meet {dom(F.0)} by
A4,XBOOLE_0:def 10;
  then dom((Partial_Sums F).n) = meet{dom(F.0)} by A1,Th28;
  hence thesis by SETFAM_1:10;
end;
