reserve PTN for Petri_net;
reserve S0 for Subset of the carrier of PTN;
reserve T0 for Subset of the carrier' of PTN;

theorem
  the carrier of PTN = the carrier of PTN.: & the carrier' of PTN = the
  carrier' of PTN.: & (the S-T_Arcs of PTN)~ = the T-S_Arcs of PTN.: & (the
  T-S_Arcs of PTN)~ = the S-T_Arcs of PTN.:;
