theorem
  *'{}the carrier' of PTN = {}
proof
  set x = the Element of *'{}the carrier' of PTN;
  assume not thesis;
  then x in *'{}the carrier' of PTN;
  then ex s being place of PTN st x = s & ex f being S-T_arc of PTN, t being
  transition of PTN st t in {}the carrier' of PTN & f = [s,t];
  hence contradiction;
end;
