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 Thc:
  dct.i in places_of dct & i in dom dct implies i mod 2 = 1
  proof
    assume that
T2: dct.i in places_of dct and
T4: i in dom dct;
     consider p be place of Dftn such that
T3:  p = dct.i & p in rng dct by T2;
E1:  1 <= i & i <= len dct by T4,FINSEQ_3:25;
E41: i = len dct or i < len dct by XXREAL_0:1,E1;
     i mod 2 = 1
     proof
       assume
E6:    i mod 2 <> 1;
       reconsider i1 = i - 1 as Element of NAT by NAT_1:21,T4, FINSEQ_3:25;
       now
         assume i1 mod 2 = 0;then
         i1 + 1 mod 2 = 1 by NAT_D:16;
         hence contradiction by E6;
       end;then
       i1 mod 2 = 1 by NAT_D:12;then
       [dct.i1,dct.(i1+1)] in the S-T_Arcs of Dftn by E6,E41,Def5;then
       dct.(i1+1) in the carrier' of Dftn by ZFMISC_1:87;
       hence contradiction by XBOOLE_0:3,NET_1:def 2,T3;
     end;
     hence thesis;
   end;
