
theorem
  ex Omega1,Omega2,Omega3,Omega4 being non empty set st
 Omega1 c< Omega2 & Omega2 c< Omega3 & Omega3 c< Omega4 &
 (ex F1 being SigmaField of Omega1,
    F2 being SigmaField of Omega2,
    F3 being SigmaField of Omega3,
    F4 being SigmaField of Omega4 st
 F1 c= F2 & F2 c= F3 & F3 c= F4)
proof
 set Omega1={1};
 set Omega2={1,2};
 set Omega3={1,2,3};
 set Omega4={1,2,3,4};
 reconsider F1 = bool Omega1 as SigmaField of Omega1;
 reconsider F2 = bool Omega2 as SigmaField of Omega2;
 reconsider F3 = bool Omega3 as SigmaField of Omega3;
 reconsider F4 = bool Omega4 as SigmaField of Omega4;
 ATh100: Omega1 c< Omega2 by ZFMISC_1:7,20;
 E10: F1 c= F2 by ZFMISC_1:7,ZFMISC_1:67;
 E11: F2 c= F3 by ATh101,ZFMISC_1:67;
 F3 c= F4 by ATh102,ZFMISC_1:67;
 hence thesis by ATh100,ATh101,ATh102,E10,E11;
end;
