theorem
  D c= X iff D in BOOL X
proof
  thus D c= X implies D in BOOL X
  proof
A1: not D in {{}} by TARSKI:def 1;
    assume D c= X;
    hence thesis by A1,XBOOLE_0:def 5;
  end;
  assume D in BOOL X;
  hence thesis;
end;
