reserve X for set;

theorem Th4:
  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 union rng G = F.0 \ meet rng F
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: dom G = NAT by FUNCT_2:def 1;
  thus union rng G c= F.0 \ meet rng F
  proof
    let A be object;
    assume A in union rng G;
    then consider Z being set such that
A4: A in Z and
A5: Z in rng G by TARSKI:def 4;
    consider n being object such that
A6: n in NAT and
A7: Z = G.n by A3,A5,FUNCT_1:def 3;
    reconsider n as Element of NAT by A6;
    consider k being Nat such that
A8: n = k + 1 by A1,A4,A7,NAT_1:6;
    reconsider k as Element of NAT by ORDINAL1:def 12;
    set Y = F.k;
A9: A in F.0 \ F.k by A2,A4,A7,A8;
    then Y in rng F & not A in Y by FUNCT_2:4,XBOOLE_0:def 5;
    then
A10: not A in meet rng F by SETFAM_1:def 1;
    A in F.0 by A9,XBOOLE_0:def 5;
    hence thesis by A10,XBOOLE_0:def 5;
  end;
  let A be object;
  assume
A11: A in F.0 \ meet rng F;
  then not A in meet rng F by XBOOLE_0:def 5;
  then
A12: ex Y being set st Y in rng F & not A in Y by SETFAM_1:def 1;
  A in F.0 by A11,XBOOLE_0:def 5;
  then consider Y being set such that
A13: A in F.0 and
A14: Y in rng F and
A15: not A in Y by A12;
  dom F = NAT by FUNCT_2:def 1;
  then consider n being object such that
A16: n in NAT and
A17: Y = F.n by A14,FUNCT_1:def 3;
  reconsider n as Element of NAT by A16;
  A in F.0 \ F.n by A13,A15,A17,XBOOLE_0:def 5;
  then
A18: A in G.(n+1) by A2;
  G.(n + 1) in rng G by FUNCT_2:4;
  hence thesis by A18,TARSKI:def 4;
end;
