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

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