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

theorem
  e_adjac(N) c= [:e_shore(N),e_shore(N):] &
  e_flow(N) c= [:e_shore(N),e_shore(N):]
proof
A1: ((the entrance of N) \/ (the escape of N)) \ id(the carrier of N) c= ((
  the entrance of N) \/ (the escape of N)) by XBOOLE_1:36;
A2: the escape of N c= [:the carrier of N,the carrier of N:] by Def1;
A3: the entrance of N c= [:the carrier of N,the carrier of N:] by Def1;
  then (the entrance of N)~ c= [:the carrier of N,the carrier of N:] by
SYSREL:4;
  then
A4: id(the carrier of N) c= [:the carrier of N,the carrier of N:] & (the
  entrance of N)~ \/ (the escape of N) c= [:the carrier of N,the carrier of N:]
  by A2,RELSET_1:13,XBOOLE_1:8;
  id((the carrier of N) \ rng(the entrance of N)) c= id(the carrier of N)
& id (the carrier of N) c= [:the carrier of N,the carrier of N:] by RELSET_1:13
,SYSREL:15,XBOOLE_1:36; then
A5: id((the carrier of N) \ rng(the entrance of N)) c= [:the carrier of N,
  the carrier of N:] by XBOOLE_1:1;
  (the entrance of N) \/ (the escape of N) c= [:the carrier of N,the
  carrier of N:] by A3,A2,XBOOLE_1:8; then
  ((the entrance of N) \/ (the escape of N)) \ id(the carrier of N) c= [:
  the carrier of N,the carrier of N:] by A1,XBOOLE_1:1;
  hence thesis by A5,A4,XBOOLE_1:8;
end;
