theorem Th43:
  (for q being State of tfsm1, s holds Tf.((the Tran of tfsm1).(q,
  s)) = (the Tran of tfsm2).(Tf.q,s)) implies for k st 1 <= k & k <= len w + 1
  holds Tf.((q11,w)-admissible.k) = (Tf.q11,w)-admissible.k
proof
  defpred P[Nat] means 1 <= $1 & $1 <= len w + 1 implies Tf.((q11,w
  )-admissible.$1) = (Tf.q11,w)-admissible.$1;
  assume
A1: for q being State of tfsm1, s holds Tf.((the Tran of tfsm1).(q,s)) =
  (the Tran of tfsm2).(Tf.q,s);
A2: for k being Nat st P[k] holds P[k + 1]
  proof
    let k be Nat;
    assume
A3: 1 <= k & k <= len w + 1 implies Tf.((q11,w)-admissible.k) = (Tf.
    q11,w)-admissible.k;
    assume that
    1<=k+1 and
A4: k+1<=len w+1;
A5: k=0 or 0<k & 0 + 1 = 1;
    per cases by A5,NAT_1:13;
    suppose
A6:   k = 0;
      hence Tf.((q11,w)-admissible.(k+1)) = Tf.q11 by Def2
        .= (Tf.q11,w)-admissible.(k+1) by A6,Def2;
    end;
    suppose
A7:   1 <= k;
A8:   len w <= len w + 1 by NAT_1:11;
A9:   k <= len w by A4,XREAL_1:6;
      then consider
      wi being Element of IAlph,qi,qi1 being State of tfsm1 such that
A10:  wi = w.k & qi = (q11,w)-admissible.k and
A11:  qi1 = (q11,w)-admissible.(k+1) & wi-succ_of qi =qi1 by A7,Def2;
      consider wri being Element of IAlph, qri, qri1 being State of tfsm2 such
      that
A12:  wri = w.k & qri = (Tf.q11,w)-admissible.k and
A13:  qri1 = (Tf.q11,w)-admissible.(k+1) & wri-succ_of qri = qri1 by A7,A9,Def2
;
      thus Tf.((q11,w)-admissible.(k+1)) = Tf.((the Tran of tfsm1).(qi, wi))
      by A11
        .= (the Tran of tfsm2).(qri, wri) by A1,A3,A7,A9,A8,A10,A12,XXREAL_0:2
        .= (Tf.q11,w)-admissible.(k+1) by A13;
    end;
  end;
A14: P[0];
  thus for k being Nat holds P[k] from NAT_1:sch 2(A14, A2);
end;
