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

theorem Th13:
  rng (the entrance of N) = rng (CL(the entrance of N)) &
  rng (the entrance of N) = dom (CL(the entrance of N)) &
  rng (the escape of N) = rng (CL(the escape of N)) &
  rng (the escape of N) = dom (CL(the escape of N)) &
  rng the entrance of N = rng the escape of N
proof
  (the entrance of N) * ((the entrance of N) \ id(the carrier of N)) = {}
  by Def2; then
A1: (the entrance of N) * ((the entrance of N) \ id(dom (the entrance of N))
  ) = {} by Th11;
  (the escape of N) * ((the escape of N) \ id(the carrier of N)) = {} by Def2;
 then
A2: (the escape of N) * ((the escape of N) \ id(dom(the escape of N))) = {}
  by Th11;
A3: (the escape of N) * (the escape of N) = the escape of N by Def1; then
A4: rng (the escape of N) = rng (CL(the escape of N)) by A2,SYSREL:31;
A5: (the entrance of N) * (the entrance of N) = the entrance of N by Def1;
  then rng (the entrance of N) = rng (CL(the entrance of N)) by A1,SYSREL:31;
  hence thesis by A5,A1,A3,A2,A4,Th12,SYSREL:31;
end;
