reserve m, n, i, k for Nat;
reserve IAlph, OAlph for non empty set,
  fsm for non empty FSM over IAlph,
  s for Element of IAlph,
  w, w1, w2 for FinSequence of IAlph,
  q, q9, q1, q2 for State of fsm;

theorem Th5:
  for k st 1 <= k & k <= len w1 holds
  (q1,w1^w2)-admissible.k = (q1,w1)-admissible.k
proof
  set q1w = (q1,w1^w2)-admissible;
  set q1w1 = (q1,w1)-admissible;
  defpred P[Nat] means 1 <= $1 & $1 <= len w1 implies q1w.$1 = q1w1
  .$1;
A1: now
    let k be Nat;
    assume
A2: P[k];
    thus P[k+1]
    proof
      assume that
      1 <= k+1 and
A3:   k+1 <= len w1;
A4:   0 = k or 0 < k & 0 + 1= 1;
      per cases by A3,A4,NAT_1:13;
      suppose
A5:     k = 0;
        hence q1w.(k+1) = q1 by Def2
          .= q1w1.(k+1) by A5,Def2;
      end;
      suppose
A6:     1 <= k & k <= len w1;
        len w1 <= len w1 + len w2 by NAT_1:11;
        then k <= len w1 + len w2 by A6,XXREAL_0:2;
        then k <= len (w1^w2) by FINSEQ_1:22;
        then
A7:     ex wk being Element of IAlph,qwk,qwk1 being State of fsm st wk = (
w1^w2).k & qwk = q1w.k & qwk1 = q1w.(k+1) & wk-succ_of qwk = qwk1 by A6,Def2;
        (ex w1k being Element of IAlph, qw1k, qw1k1 being State of fsm st
w1k = w1.k & qw1k = q1w1.k & qw1k1 = q1w1.(k+1) & w1k-succ_of qw1k = qw1k1 )& k
        in dom w1 by A6,Def2,FINSEQ_3:25;
        hence thesis by A2,A6,A7,FINSEQ_1:def 7;
      end;
    end;
  end;
A8: P[0];
  thus for k being Nat holds P[k] from NAT_1:sch 2 (A8,A1 );
end;
