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

theorem
  for N being Pnet for y being Element of Elements(N) holds y in the
  carrier' of N implies (x in Pre(N,y) iff pre N,y,x)
proof
  let N be Pnet;
  let y be Element of Elements(N);
  assume
A1: y in the carrier' of N;
A2: pre N,y,x implies x in Pre(N,y)
  proof
    assume pre N,y,x;
    then
A3: [x,y] in Flow N;
    then x in the carrier of N by A1,Th6;
    then x in Elements(N) by XBOOLE_0:def 3;
    hence thesis by A3,Def6;
  end;
  x in Pre(N,y) implies pre N,y,x
  by Def6,A1;
  hence thesis by A2;
end;
