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

theorem Th15:
  (the entrance of N) * ((the escape of N) \ id the carrier of N) = {} &
  (the escape 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;
A1: S * (R \ T) = S * (R \ id dom R) by Th11
    .= (S * R) * (R \ id dom R) by Def1
    .= S * (R * (R \ id dom R)) by RELAT_1:36
    .= S * (R * (R \ T)) by Th11
    .= S * {} by Def2
    .= {};
  R * (S \ T) = R * (S \ id dom S) by Th11
    .= (R * S) * (S \ id dom S) by Def1
    .= R * (S * (S \ id dom S)) by RELAT_1:36
    .= R * (S * (S \ T)) by Th11
    .= R * {} by Def2
    .= {};
  hence thesis by A1;
end;
