reserve Omega for set;
reserve X, Y, Z, p,x,y,z for set;
reserve D, E for Subset of Omega;
reserve f for Function;
reserve m,n for Nat;
reserve r,r1 for Real;
reserve seq for Real_Sequence;
reserve F for Field_Subset of X;
reserve ASeq,BSeq for SetSequence of Omega;
reserve A1 for SetSequence of X;

theorem Th16:
  Y is SigmaField of X implies Y is Field_Subset of X
proof
  assume
A1: Y is SigmaField of X;
  Y is cap-closed
  proof
    let A,B be set;
    assume
A2: A in Y & B in Y;
    then reconsider A9 = A, B9 = B as Subset of X by A1;
    set A1 = A9 followed_by B9;
    rng A1 = {A9,B9} by FUNCT_7:126;
    then
A3: rng A1 c= Y by A2,ZFMISC_1:32;
    Intersection A1 = A /\ B by Th14;
    hence thesis by A1,A3,Def6;
  end;
  hence thesis by A1;
end;
