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;
reserve Dftn for With_directed_path Petri Petri_net,
  dct for directed_path_like FinSequence of places_and_trans_of Dftn;

theorem Thf:
  dct.i in transitions_of dct & i in dom dct implies
  [dct.(i-1),dct.i] in the S-T_Arcs of Dftn
  & [dct.i,dct.(i+1)] in the T-S_Arcs of Dftn
  proof
    assume
H1: dct.i in transitions_of dct & i in dom dct;
L1:  [dct.(len dct - 1),dct.len dct] in the T-S_Arcs of Dftn by The;
     reconsider im1 = i - 1 as Element of NAT by NAT_1:21,FINSEQ_3:25,H1;
     consider t be transition of Dftn such that
H6:  t = dct.i & t in rng dct by H1;
H4:  1 <= i & i <= len dct by FINSEQ_3:25,H1;
     now
       assume im1 mod 2 = 0;then
       im1 + 1 mod 2 = 1 by NAT_D:16;
       hence contradiction by Thcc,H1;
     end;then
H2:  im1 mod 2 = 1 by NAT_D:12;
     i <> len dct
     proof
       assume i = len dct;then
       dct.i in the carrier of Dftn by L1,ZFMISC_1:87;
       hence contradiction by H6,NET_1:def 2 ,XBOOLE_0:3;
     end;then
H3:  im1 + 1 < len dct by XXREAL_0:1,H4;
     hence [dct.(i-1),dct.i] in the S-T_Arcs of Dftn by Def5,H2;
     [dct.(im1+1),dct.(im1+2)] in the T-S_Arcs of Dftn by Def5,H2,H3;
     hence thesis;
   end;
