theorem
  for Z being Field_Subset of Omega holds sigma Z = monotoneclass Z
proof
  let Z be Field_Subset of Omega;
  monotoneclass Z is Field_Subset of Omega by Th72;
  then
A1: monotoneclass Z is SigmaField of Omega by Th70;
  Z c= monotoneclass Z by Def9;
  hence sigma Z c= monotoneclass Z by A1,PROB_1:def 9;
  sigma Z is MonotoneClass of Omega & Z c= sigma Z by Th70,PROB_1:def 9;
  hence thesis by Def9;
end;
