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

theorem
  ((the escape of N) \ id(the carrier of N))~ * id((the carrier of N) \
    rng(the escape of N))~ = ((the escape of N) \ id(the carrier of N))~ &
  ((the entrance of N) \ id(the carrier of N))~ *
    id((the carrier of N) \ rng(the entrance of N))~ =
      ((the entrance of N) \ id(the carrier of N))~
proof
A1: ((the entrance of N) \ id(the carrier of N))~ * id((the carrier of N) \
rng(the entrance of N))~ = ((id((the carrier of N) \ rng(the entrance of N))) *
  ((the entrance of N) \ id(the carrier of N)))~ by RELAT_1:35
    .= ((the entrance of N) \ id(the carrier of N))~ by Th23;
  ((the escape of N) \ id(the carrier of N))~ * id((the carrier of N) \
rng(the escape of N))~ = ((id((the carrier of N) \ rng(the escape of N))) * ((
  the escape of N) \ id(the carrier of N)))~ by RELAT_1:35
    .= ((the escape of N) \ id(the carrier of N))~ by Th23;
  hence thesis by A1;
end;
