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;
reserve M0 for marking of PTN,
       t for transition of PTN,
       Q,Q1 for FinSequence of the carrier' of PTN;

theorem
  for Dftn being Decision_free_PT,
  dct being Circuit_of_places_and_trans of Dftn,
  M0 being marking of Dftn, Q being FinSequence of the carrier' of Dftn holds
  num_marks(places_of dct, M0) = num_marks(places_of dct, Firing(Q, M0))
  proof
    let Dftn be With_directed_circuit Petri Decision_free_PT,
    dct be Circuit_of_places_and_trans of Dftn,
    M0 be marking of Dftn,
    Q be FinSequence of the carrier' of Dftn;
    set P = places_of dct, F = Firing(Q, M0);
    per cases;
    suppose
C1:   Q <> {};then
      consider M being FinSequence of nat_marks_of Dftn such that
A2:   len Q = len M and
A2a:  F = M/.len M and
A2b:  M/.1 = Firing(Q/.1, M0) and
A2c:  for i st i < len Q & i > 0
      holds M/.(i+1) = Firing(Q/.(i+1),M/.i) by Defb;
      defpred R[Nat] means 1<=$1 & $1 <= len Q implies
      num_marks(P,M/.1) = num_marks(P,M/.$1);
A5:   R[0];
A4:   now
        let i;
        assume
A10:    R[i];
        thus R[i+1]
        proof
          assume
A9:       1 <= i+1 & i+1 <= len Q;then
X1:       i < len Q by NAT_1:13;
          per cases;
          suppose 0 = i;
            hence num_marks(P,M/.1) = num_marks(P,M/.(i+1));
          end;
          suppose
S2:         0 < i;then
            0 + 1 <= i by NAT_1:13;
            hence num_marks(P,M/.1)
             = num_marks(P,Firing(Q/.(i+1),M/.i)) by Th7,A10,NAT_1:13,A9
            .= num_marks(P,M/.(i+1)) by A2c,S2,X1;
          end;
        end;
      end;
A6:   for i holds R[i] from NAT_1:sch 2(A5,A4);
      0+1 <= len Q by NAT_1:13,C1;then
      num_marks(P,M/.1) = num_marks(P,M/.len M) by A2, A6;
      hence num_marks(P, M0) = num_marks(P, Firing(Q, M0)) by A2a,A2b,Th7;
    end;
    suppose Q = {};
      hence thesis by Defb;
    end;
  end;
