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

theorem
  f_prox(M) c= [:Elements(M), Elements(M):] &
  f_flow(M) c= [:Elements(M), Elements(M):]
proof
A1: (Flow M)|(the carrier of M) c= Flow M by RELAT_1:59;
  Flow M c= [:Elements(M), Elements(M):] by Th8;
  then
A2: (Flow M)|(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
A4: (Flow M)~|(the carrier of M) c= [:Elements(M), Elements(M):]
  by A3,XBOOLE_1:1;
  the carrier of M c= Elements(M) by XBOOLE_1:7;
  then
A5: [:the carrier of M, the carrier of M:] c=
  [:Elements(M), Elements(M):] by ZFMISC_1:96;
  id(the carrier of M) c= [:the carrier of M, the carrier of M:]
  by RELSET_1:13;
  then
A6: id(the carrier of M) c= [:Elements(M), Elements(M):] by A5,XBOOLE_1:1;
  (Flow M)|(the carrier of M) \/
  (Flow M)~|(the carrier of M) c= [:Elements(M), Elements(M):]
  by A2,A4,XBOOLE_1:8;
  hence f_prox(M) c= [:Elements(M), Elements(M):] by A6,XBOOLE_1:8;
A7: Flow M c= [:Elements(M), Elements(M):] by Th8;
  id(Elements(M)) c= [:Elements(M), Elements(M):] by RELSET_1:13;
  hence thesis by A7,XBOOLE_1:8;
end;
