reserve X for set;

theorem Th5:
  for S being SigmaField of X, G,F being sequence of S st (G.0
= {} & for n being Nat holds G.(n+1) = F.0 \ F.n & F.(n+1) c= F.n )
  holds meet rng F = F.0 \ union rng G
proof
  let S be SigmaField of X;
  let G,F be sequence of S;
  assume that
A1: G.0 = {} and
A2: for n being Nat holds G.(n+1) = F.0 \ F.n & F.(n+1) c= F .n;
A3: for n being Nat holds F.n c= F.0
  proof
    defpred P[Nat] means F.$1 c= F.0;
A4: for k being Nat st P[k] holds P[k+1]
    proof
      let k be Nat;
      assume
A5:   F.k c= F.0;
      F.(k+1) c= F.k by A2;
      hence thesis by A5,XBOOLE_1:1;
    end;
A6: P[0];
    thus for n being Nat holds P[n] from NAT_1:sch 2(A6,A4);
  end;
A7: meet rng F c= F.0
  proof
    set X = the Element of rng F;
    let A be object;
    dom F = NAT by FUNCT_2:def 1;
    then ex n being object st n in NAT & F.n = X by FUNCT_1:def 3;
    then
A8: X c= F.0 by A3;
    assume A in meet rng F;
    then A in X by SETFAM_1:def 1;
    hence thesis by A8;
  end;
A9: F.0 /\ meet rng F = F.0 \ (F.0 \ meet rng F) by XBOOLE_1:48;
  union rng G = F.0 \ meet rng F by A1,A2,Th4;
  hence thesis by A7,A9,XBOOLE_1:28;
end;
