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 Th7:
  for x,y being Subset of Omega st x misses y holds (x,y)
  followed_by {} Omega is disjoint_valued
proof
  let x,y be Subset of Omega such that
A1: x misses y;
  reconsider r=(x,y) followed_by {} Omega as sequence of  bool Omega;
  now
    let n,m;
A2: m>1 implies r.m={} by FUNCT_7:124;
    assume
A3: n<m;
A4: now
      assume
A5:   m=0 or m=1;
      0+1=1;
      then n <= 0 by A3,A5,NAT_1:13;
      hence n=0 & m=1 by A3,A5,NAT_1:3;
    end;
    r.0=x by FUNCT_7:122;
    hence r.n misses r.m by A1,A4,A2,FUNCT_7:123,NAT_1:25,XBOOLE_1:65;
  end;
  hence thesis;
end;
