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 Thcc:
  dct.i in transitions_of dct & i in dom dct implies i mod 2 = 0
  proof
    assume that
T2: dct.i in transitions_of dct and
T4: i in dom dct;
L1: [dct.(len dct - 1),dct.len dct] in the T-S_Arcs of Dftn by The;
    consider t be transition of Dftn such that
T3: t = dct.i & t in rng dct by T2;
T5: 1 <= i & i <= len dct by T4,FINSEQ_3:25;
    i <> len dct
    proof
      assume i = len dct;then
      dct.i in the carrier of Dftn by L1,ZFMISC_1:87;
      hence contradiction by NET_1:def 2,XBOOLE_0:3,T3;
    end;then
    i < len dct by XXREAL_0:1,T5;then
H4: i+1 <= len dct by NAT_1:13;
    assume i mod 2 <> 0;then
H1: i mod 2 = 2-1 by NAT_D:12;
    i+1<>len dct
    proof
      assume
H6:   i+1 = len dct;then
      reconsider p = dct.(i+1) as place of Dftn by L1,ZFMISC_1:87;
      1 <= 1+i by NAT_1:11;then
H5:   i+1 in dom dct by FINSEQ_3:25,H6;then
      p in rng dct by FUNCT_1:3;then
      p in places_of dct;then
      i+1 mod 2 = 1 by Thc,H5;
      hence contradiction by NAT_D:69, H1;
    end;then
    i + 1 < len dct by XXREAL_0:1,H4;then
    [dct.i,dct.(i+1)] in the S-T_Arcs of Dftn by Def5,H1;then
    dct.i in the carrier of Dftn by ZFMISC_1:87;
    hence contradiction by XBOOLE_0:3,T3,NET_1:def 2;
  end;
