reserve x,y for object,X,Y for set;
reserve M for Pnet;

theorem
  f_exit(M) c= [:Elements(M),Elements(M):] &
  f_enter(M) c= [:Elements(M),Elements(M):]
proof
A1: id(the carrier of M) c= id(Elements(M)) by SYSREL:15,XBOOLE_1:7;
  id(Elements(M)) c= [:Elements(M),Elements(M):] by RELSET_1:13;
  then
A2: id(the carrier of M) c= [:Elements(M),Elements(M):] by A1,XBOOLE_1:1;
A3: (Flow M)|(the carrier' of M) c= (Flow M) by RELAT_1:59;
  (Flow M) c= [:Elements(M),Elements(M):] by Th8;
  then (Flow M)|(the carrier' of M) c= [:Elements(M),Elements(M):]
  by A3,XBOOLE_1:1;
  hence f_exit(M) c= [:Elements(M),Elements(M):] by A2,XBOOLE_1:8;
A4: id(the carrier of M) c= id(Elements(M)) by SYSREL:15,XBOOLE_1:7;
  id(Elements(M)) c= [:Elements(M),Elements(M):] by RELSET_1:13;
  then
A5: id(the carrier of M) c= [:Elements(M),Elements(M):] by A4,XBOOLE_1:1;
  A6: ( Flow M)~|(the carrier' of M) c= (Flow M)~ by RELAT_1:59;
  (Flow M)~ c= [:Elements(M),Elements(M):] by Th8;
  then (Flow M)~|(the carrier' of M) c= [:Elements(M),Elements(M):]
  by A6,XBOOLE_1:1;
  hence thesis by A5,XBOOLE_1:8;
end;
