reserve N for PT_net_Str, PTN for Petri_net, i for Nat;
reserve fs for FinSequence of places_and_trans_of PTN;

theorem Th12:
  for fs be FinSequence of places_and_trans_of PetriNet st fs = <*0,1,0*>
  holds fs is directed_path_like
  proof
    let fs be FinSequence of places_and_trans_of PetriNet;
    assume
L1: fs = <*0,1,0*>;
A12: fs.1 = 0 & fs.2 = 1 & fs.3 = 0 by L1;
     set N = PetriNet;
     thus fs is directed_path_like
     proof
       2 mod 2 = 0 by NAT_D:25;then
L3:    2 + 1 mod 2 = 1 by NAT_D:16;
L4:    now
         let i;
         assume
A16:     i mod 2 = 1 & i + 1 < len fs;
         0 mod 2 = 0 by NAT_D:26;then
         0 < i by A16;then
A17:     0+1 <= i by NAT_1:13;
         now
           assume 1 < i;then
           1+1<=i by NAT_1:13;then
           2+1<=i+1 by XREAL_1:6;
           hence contradiction by A16,FINSEQ_1:45, L1;
         end;then
A11:     i = 1 by XXREAL_0:1,A17;
A15:     0 in {0} & 1 in {1} by TARSKI:def 1;
         thus [fs.i, fs.(i+1)] in (the S-T_Arcs of N) &
         [fs.(i+1),fs.(i+2)] in (the T-S_Arcs of N)
         by A11, A12,ZFMISC_1:def 2, A15;
       end;
       fs.len fs = 0 by A12,FINSEQ_1:45, L1;
       hence thesis by FINSEQ_1:45,L1,L3,L4,TARSKI:def 1;
     end;
   end;
