reserve Omega, F for non empty set,
  f for SetSequence of Omega,
  X,A,B for Subset of Omega,
  D for non empty Subset-Family of Omega,
  n,m for Element of NAT,
  h,x,y,z,u,v,Y,I for set;

theorem Th15:
  D is Dynkin_System of Omega & D is intersection_stable implies
for f being SetSequence of Omega holds rng f c= D implies rng disjointify(f) c=
  D
proof
  assume
A1: D is Dynkin_System of Omega & D is intersection_stable;
  let f be SetSequence of Omega;
  assume
A2: rng f c= D;
A3: for n holds (disjointify(f)).n in D
  proof
    let n;
A4: rng (f|n) c= rng(f) by RELAT_1:70;
A5: union rng(f|n)in D by A1,A2,A4,Th14,XBOOLE_1:1;
    then reconsider urf=union rng(f|n) as Subset of Omega;
    dom(f)=NAT by FUNCT_2:def 1;
    then f.n in rng f by FUNCT_1:def 3;
    then f.n \ urf in D by A1,A2,A5,Th12;
    hence thesis by Th4;
  end;
    let y be object;
    assume y in rng disjointify(f);
    then consider x being object such that
A6: x in dom(disjointify(f)) and
A7: y=(disjointify(f)).x by FUNCT_1:def 3;
    reconsider n=x as Element of NAT by A6,FUNCT_2:def 1;
    y=(disjointify(f)).n by A7;
    hence y in D by A3;
end;
