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 Th14:
  D is Dynkin_System of Omega & D is intersection_stable implies
  for x being finite set holds x c= D implies union x in D
proof
  assume that
A1: D is Dynkin_System of Omega and
A2: D is intersection_stable;
  defpred P[set] means union $1 in D;
  let x be finite set;
  assume
A3: x c= D;
A4: for y,b being set st y in x & b c= x & P[b] holds P[b \/ {y}]
  proof
    let y,b be set such that
A5: y in x and
    b c= x and
A6: union b in D;
    y in D by A3,A5;
    then reconsider y1=y as Subset of Omega;
    reconsider unionb = union b as Subset of Omega by A6;
    union {y}=y & unionb \/ y1 in D by A1,A2,A3,A5,A6,Th13,ZFMISC_1:25;
    hence thesis by ZFMISC_1:78;
  end;
A7: x is finite;
A8: P[{}] by A1,Def5,ZFMISC_1:2;
  thus P[x] from FINSET_1:sch 2(A7,A8,A4);
end;
