theorem Th17:
  qa9 = (the Tran of tfsm).[qa, s] implies for i st i in Seg (len
  w + 1) holds (qa, <*s*>^w)-admissible.(i+1) = (qa9, w)-admissible.i
proof
  set sw = (<*s*>^w);
A1: len sw = len <*s*> + len w by FINSEQ_1:22
    .= len w + 1 by FINSEQ_1:40;
  defpred P[Nat] means $1 in Seg (len w + 1) implies (qa, <*s*>^w)
  -admissible.($1+1) = (qa9, w)-admissible.$1;
A2: sw.1 = s by FINSEQ_1:41;
  assume
A3: qa9 = (the Tran of tfsm).[qa, s];
A4: for j being Nat st P[j] holds P[j+1]
  proof
    let j be Nat;
    assume
A5: j in Seg (len w + 1) implies (qa, <*s*>^w)-admissible.(j+1) = (qa9
    , w)-admissible.j;
    assume
A6: j+1 in Seg (len w + 1);
    per cases;
    suppose
A7:   j = 0;
      set aadm = (qa, sw)-admissible;
      1 <= len sw by A1,A6,A7,FINSEQ_1:1;
      then
A8:   ex swi1 being Element of IAlph, a1, a11 being Element of tfsm st
      swi1 = sw.1 & a1 = aadm.1 & a11 = aadm.(1+1) & swi1-succ_of a1 = a11 by
Def2;
      (qa9, w)-admissible.1 = qa9 by Def2;
      hence thesis by A3,A2,A7,A8,Def2;
    end;
    suppose
A9:   j <> 0;
      set aadm = (qa, sw)-admissible, aadm9 = (qa9, w)-admissible;
A10:  j in Seg len w by A6,A9,FINSEQ_1:61;
      then
A11:  j <= len w by FINSEQ_1:1;
      then 1 <= j+1 & j+1 <= len sw by A1,NAT_1:12,XREAL_1:7;
      then
A12:  ex swj1 being Element of IAlph, aj1, aj11 being Element of tfsm st
swj1 = sw.(j+1) & aj1 = aadm.(j+1) & aj11 = aadm.(j+1+1) & swj1 -succ_of aj1 =
      aj11 by Def2;
      1 <= j by A10,FINSEQ_1:1;
      then
A13:  ex wj being Element of IAlph, aj9, aj19 being Element of tfsm st wj
= w.j & aj9 = aadm9.j & aj19 = aadm9.(j+1) & wj-succ_of aj9 = aj19 by A11,Def2;
      j in dom w by A10,FINSEQ_1:def 3;
      hence thesis by A5,A6,A9,A12,A13,FINSEQ_1:61,FINSEQ_3:103;
    end;
  end;
A14: P[0] by FINSEQ_1:1;
  thus for i being Nat holds P[i] from NAT_1:sch 2(A14,A4);
end;
