reserve N for PT_net_Str, PTN for Petri_net, i for Nat;
reserve fs for FinSequence of places_and_trans_of PTN;
 reserve Dftn for With_directed_path Petri_net;
 reserve dct for directed_path_like FinSequence of places_and_trans_of Dftn;

theorem The:
  [dct.(len dct -1), dct.(len dct)] in the T-S_Arcs of Dftn
  proof
    len dct >= 3 by Def5;then
    reconsider ln2 = len dct - 2 as Element of NAT by NAT_1:21, XXREAL_0:2;
F8: 1 = (ln2 + 2) mod 2 by Def5
    .= ((ln2 mod 2) + (2 mod 2)) mod 2 by NAT_D:66
    .= ((ln2 mod 2) + 0) mod 2 by NAT_D:25
    .= ln2 mod 2 by NAT_D:65;
    len dct + (-1) < len dct by XREAL_1:30; then
    [dct.(ln2 + 1), dct.(ln2 + 2)] in the T-S_Arcs of Dftn by Def5, F8;
    hence thesis;
  end;
