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

theorem
  e_Flow N c= [:e_Places(N), e_Transitions(N):] \/
    [:e_Transitions(N), e_Places(N):]
proof
A1: (the entrance of N)~ \ id N = (the entrance of N)~ \ (id
  (the carrier of N))~
    .= ((the entrance of N) \ id(the carrier of N))~ by RELAT_1:24;
A2: e_Flow(N) = ((the entrance of N)~ \ id(the carrier of N)) \/ ((the
escape of N) \ id(the carrier of N)) & (the escape of N) \ id(the carrier of N)
  c= [: e_Transitions(N) , e_Places(N):] by Th18,XBOOLE_1:42;
  (the entrance of N) \ id(the carrier of N) c= [:e_Transitions(N) ,
  e_Places(N):] by Th18;
  then (the entrance of N)~ \ id(the carrier of N) c= [:e_Places(N) ,
  e_Transitions(N):] by A1,SYSREL:4;
  hence thesis by A2,XBOOLE_1:13;
end;
