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 Thg:
  dct.i in places_of dct & 1 < i & i < len dct implies
  [dct.(i-2),dct.(i-1)] in the S-T_Arcs of Dftn
  & [dct.(i-1),dct.i] in the T-S_Arcs of Dftn
  & [dct.i,dct.(i+1)] in the S-T_Arcs of Dftn
  & [dct.(i+1),dct.(i+2)] in the T-S_Arcs of Dftn & 3 <= i
  proof
    assume
H1: dct.i in places_of dct & 1 < i & i < len dct; then
P1: i in dom dct by FINSEQ_3:25;then
H4: i mod 2 = 1 by Thc,H1;
L1: [dct.(len dct - 1),dct.len dct] in the T-S_Arcs of Dftn by The;
L2: [dct.1,dct.2] in the S-T_Arcs of Dftn by Thd;
    consider p be place of Dftn such that
H6: p = dct.i & p in rng dct by H1;
H8: 1+1 <= i by H1,NAT_1:13;then
    reconsider im2 = i - 2 as Element of NAT by NAT_1:21;
    now
      assume im2 mod 2 = 0;then
      im2 + 1 mod 2 = 1 mod 2 by NAT_D:17 .= 2 - 1 by NAT_D:14;then
      im2 + 1 + 1 mod 2 = 0 by NAT_D:69;
      hence contradiction by Thc,H1, P1;
    end;then
H2: im2 mod 2 = 1 by NAT_D:12;
P2: i - 1 < len dct by H1,XREAL_1:147;then
    [dct.im2,dct.(im2+1)] in the S-T_Arcs of Dftn by Def5,H2;
    hence [dct.(i-2),dct.(i-1)] in the S-T_Arcs of Dftn;
    [dct.(im2+1),dct.(im2+2)] in the T-S_Arcs of Dftn by Def5,H2,P2;
    hence [dct.(i-1),dct.i] in the T-S_Arcs of Dftn;
H9: i+1<= len dct by NAT_1:13,H1;
    i + 1 <> len dct
    proof
      assume i + 1 = len dct;then
      dct.(i+1-1) in the carrier' of Dftn by L1,ZFMISC_1:87;
      hence contradiction by NET_1:def 2,XBOOLE_0:3,H6;
    end;then
H5: i+1<len dct by XXREAL_0:1,H9;
    hence [dct.i,dct.(i+1)] in the S-T_Arcs of Dftn by Def5,H4;
    thus [dct.(i+1),dct.(i+2)] in the T-S_Arcs of Dftn by Def5,H5,H4;
    2 <> i
    proof
      assume i = 2;then
      dct.i in the carrier' of Dftn by L2,ZFMISC_1:87;
      hence contradiction by NET_1:def 2,H6,XBOOLE_0:3;
    end;then
    2 < i by XXREAL_0:1,H8;then
    2+1<=i by NAT_1:13;
    hence 3 <= i;
  end;
