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

theorem
  f_pre(M) c= [:f_transitions(M),f_places(M):] &
  f_post(M) c= [:f_transitions(M),f_places(M):]
proof
A1: for x,y being object holds [x,y] in f_pre(M) implies
  [x,y] in [:f_transitions(M),f_places(M):]
  proof
    let x,y be object;
    assume
A2: [x,y] in f_pre(M);
    then
A3: x in (the carrier' of M) by RELAT_1:def 11;
    [x,y] in (Flow M) by A2,RELAT_1:def 11;
    then y in (the carrier of M) by A3,Th7;
    hence thesis by A3,ZFMISC_1:87;
  end;
  for x,y being object holds [x,y] in f_post(M) implies
  [x,y] in [:f_transitions(M),f_places(M):]
  proof
    let x,y be object;
    assume
A4: [x,y] in f_post(M);
    then
A5: [x,y] in (Flow M)~ by RELAT_1:def 11;
A6: x in (the carrier' of M) by A4,RELAT_1:def 11;
    [y,x] in (Flow M) by A5,RELAT_1:def 7;
    then y in (the carrier of M) by A6,Th7;
    hence thesis by A6,ZFMISC_1:87;
  end;
  hence thesis by A1,RELAT_1:def 3;
end;
