reserve x,y,z for object,X,Y for set;
reserve N for e_net;

theorem
  ((the entrance of N) \ id(the carrier of N)) *
    ((the entrance of N) \ id(the carrier of N)) = {} &
  ((the escape of N) \ id(the carrier of N)) *
    ((the escape of N) \ id(the carrier of N)) = {} &
  ((the entrance of N) \ id(the carrier of N)) *
    ((the escape of N) \ id(the carrier of N)) = {} &
  ((the escape of N) \ id(the carrier of N)) *
    ((the entrance of N) \ id(the carrier of N)) = {}
proof
  set R = the entrance of N;
  set S = the escape of N;
  set T = id(the carrier of N);
  (R \ T) * (S \ T) c= R * (S \ T) by RELAT_1:30,XBOOLE_1:36; then
A1: (R \ T) * (S \ T) c= {} by Th15;
  (S \ T) * (S \ T) c= S * (S \ T) by RELAT_1:30,XBOOLE_1:36; then
A2: (S \ T) * (S \ T) c= {} by Def2;
  (S \ T) * (R \ T) c= S * (R \ T) by RELAT_1:30,XBOOLE_1:36; then
A3: (S \ T) * (R \ T) c= {} by Th15;
  (R \ T) * (R \ T) c= R * (R \ T) by RELAT_1:30,XBOOLE_1:36;
  then ((R \ T) * (R \ T)) c= {} by Def2;
  hence thesis by A1,A2,A3,XBOOLE_1:3;
end;
