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;
reserve tfsm, tfsm1, tfsm2, tfsm3 for non empty Mealy-FSM over IAlph, OAlph,
  sfsm for non empty Moore-FSM over IAlph, OAlph,
  qs for State of sfsm,
  q, q1, q2 , q3, qa, qb, qc, qa9, qt, q1t, q2t for State of tfsm,
  q11, q12 for State of tfsm1,
  q21, q22 for State of tfsm2;
reserve OAlphf for finite non empty set,
  tfsmf for finite non empty Mealy-FSM over IAlph, OAlphf,
  sfsmf for finite non empty Moore-FSM over IAlph, OAlphf;
reserve tfsm, rtfsm for finite non empty Mealy-FSM over IAlph, OAlph,
  q for State of tfsm;
reserve qr1, qr2 for State of rtfsm,
  Tf for Function of the carrier of tfsm1, the carrier of tfsm2;
reserve Rtfsm for reduced finite non empty Mealy-FSM over IAlph, OAlph;
reserve Ctfsm, Ctfsm1, Ctfsm2 for connected finite non empty Mealy-FSM over
  IAlph, OAlph;

theorem Th49:
  (the Tran of tfsm)|[:accessibleStates tfsm, IAlph:] is Function
  of [:accessibleStates tfsm, IAlph:], accessibleStates tfsm
proof
  set cTran = (the Tran of tfsm) | [:accessibleStates tfsm, IAlph:];
A1: accessibleStates tfsm c= the carrier of tfsm by Th48;
  then [:accessibleStates tfsm, IAlph:] c= [:the carrier of tfsm, IAlph:] by
ZFMISC_1:96;
  then cTran is Function of [:accessibleStates tfsm, IAlph:], the carrier of
  tfsm by FUNCT_2:32;
  then
A2: dom cTran = [:accessibleStates tfsm, IAlph:] by FUNCT_2:def 1;
  rng cTran c= accessibleStates tfsm
  proof
    set I = the InitS of tfsm;
    let x be object;
    assume
A3: x in rng cTran;
    then consider d being object such that
A4: d in dom cTran and
A5: x = cTran.d by FUNCT_1:def 3;
A6: d`1 in accessibleStates tfsm by A2,A4,MCART_1:10;
    then reconsider q = d`1 as State of tfsm by A1;
    reconsider s = d`2 as Element of IAlph by A2,A4,MCART_1:10;
    set qsa = (q, <*s*>)-admissible;
A7: qsa.1 = q & <*s*>.1 = s by Def2;
    rng cTran c= the carrier of tfsm by RELAT_1:def 19;
    then reconsider q1=x as State of tfsm by A3;
    1 <= len <*s*> by FINSEQ_1:39;
    then
A8: ex wi being Element of IAlph, qi, qi1 being State of tfsm st wi=<*s*>
    .1 & qi=qsa.1 & qi1=qsa.(1+1) & wi-succ_of qi = qi1 by Def2;
    (the Tran of tfsm).d = q1 by A2,A4,A5,FUNCT_1:49;
    then
A9: qsa.(1+1) = q1 by A4,A7,A8,MCART_1:21;
    1+1 =2;
    then
A10: 2 <= len <*s*> + 1 by FINSEQ_1:39;
    q is accessible by A6,Th48;
    then consider w being FinSequence of IAlph such that
A11: I, w-leads_to q;
    len (w^<*s*>) = len w + len <*s*> by FINSEQ_1:22;
    then len(w^<*s*>)+1 =len w +1+1 by FINSEQ_1:39
      .= len w +(1+1);
    then (I, w^<*s*>)-admissible.(len (w^<*s*>) + 1) = q1 by A11,A9,A10,Th7;
    then I,w^<*s*>-leads_to q1;
    then q1 is accessible;
    hence thesis;
  end;
  hence thesis by A2,FUNCT_2:def 1,RELSET_1:4;
end;
