reserve x,y,z,a,b,c,X,A for set;
reserve C,D for Coherence_Space;

theorem Th2:
  bool X is Coherence_Space
proof
A1: for A st A c= bool X & (for a,b st a in A & b in A holds a \/ b in bool
  X) holds union A in bool X
  proof
    let A;
    assume that
A2: A c= bool X and
    for a,b st a in A & b in A holds a \/ b in bool X;
    for a st a in A holds a c= X by A2;
    then union A c= X by ZFMISC_1:76;
    hence thesis;
  end;
  for a, b st a in bool X & b c= a holds b in bool X
  proof
    let a,b;
    assume a in bool X & b c= a;
    then b c= X by XBOOLE_1:1;
    hence thesis;
  end;
  hence thesis by A1,Def1,CLASSES1:def 1;
end;
