reserve x,y for object,X,Y for set;
reserve M for Pnet;

theorem
  (f_exit(M)) * (f_exit(M) \ id(Elements(M))) = {} &
  (f_enter(M)) * (f_enter(M) \ id(Elements(M))) = {}
proof
  set S = id(the carrier of M);
  thus (f_exit(M)) * (f_exit(M) \ id(Elements(M))) = {}
  proof
    set R = (Flow M)|(the carrier' of M);
A1: S * R = {} by Th12;
    (f_exit(M)) * (f_exit(M) \ id(Elements(M))) = (R \/ S) * R by Th14
      .= (R * R) \/ (S * R) by SYSREL:6
      .= {} by A1,Th11;
    hence thesis;
  end;
  set R = ((Flow M)~|(the carrier' of M));
A2: S * R = {} by Th12;
  (f_enter(M)) * (f_enter(M) \ id(Elements(M))) = (R \/ S) * R by Th14
    .= (R * R) \/ (S * R) by SYSREL:6
    .= {} by A2,Th11;
  hence thesis;
end;
