
theorem Th5:
  for X be non empty set, A be set, S be SigmaField of X, F be
Finite_Sep_Sequence of S, G be FinSequence of S st dom G = dom F & for i be Nat
  st i in dom G holds G.i = A /\ F.i holds G is Finite_Sep_Sequence of S
proof
  let X be non empty set, A be set, S be SigmaField of X, F be
  Finite_Sep_Sequence of S, G be FinSequence of S such that
A1: dom G = dom F and
A2: for i be Nat st i in dom G holds G.i= A /\ F.i;
  now
    let i,j be Nat;
    assume that
A3: i in dom G and
A4: j in dom G and
A5: i <> j;
A6: F.i misses F.j by A1,A3,A4,A5,Th4;
    (A /\ F.i) /\ (A /\ F.j) =A /\ (F.i /\ (A /\ F.j)) by XBOOLE_1:16
      .=A /\ ((F.i /\ F.j) /\ A) by XBOOLE_1:16
      .=A /\ ({} /\ A) by A6
      .={};
    then A /\ F.i misses A /\ F.j;
    then G.i misses A /\ F.j by A2,A3;
    hence G.i misses G.j by A2,A4;
  end;
  hence thesis by Th4;
end;
