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 Th17:
  for D being Dynkin_System of Omega for x,y being Element of D
  holds x misses y implies x \/ y in D
proof
  let D be Dynkin_System of Omega;
  reconsider e={} as Element of D by Def5;
  let x,y be Element of D;
  reconsider x1=x as Subset of Omega;
  reconsider y1=y as Subset of Omega;
  reconsider r= (x1,y1) followed_by {} Omega as SetSequence of Omega;
  (x,y) followed_by e is sequence of D by Lm1;
  then
A1: rng r c= D by RELAT_1:def 19;
  assume x misses y;
  then r is disjoint_valued by Th7;
  then Union r in D by A1,Def5;
  hence thesis by Th2;
end;
