reserve x,y for set;
reserve N for PT_net_Str;

theorem Th16:
  for N being Pnet for x being Element of Elements(N) holds
  Elements(N) <> {} implies enter(N,x) c= Elements(N)
proof
  let N be Pnet;
  let x be Element of Elements(N);
  assume
A1: Elements(N) <> {};
A2: enter(N,x) ={x} implies enter(N,x) c= Elements(N)
  proof
    x in Elements(N) by A1;
    then
A3: for y being object holds y in {x} implies y in Elements(N)
        by TARSKI:def 1;
    assume enter(N,x) ={x};
    hence thesis by A3,TARSKI:def 3;
  end;
  enter(N,x) ={x} or enter(N,x) = Pre(N,x) by A1,Th15;
  hence thesis by A2,Th9;
end;
