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;

theorem Th47:
  tfsm is reduced iff ex M being finite non empty Mealy-FSM over
  IAlph,OAlph st tfsm, the_reduction_of M-are_isomorphic
proof
  set M = tfsm;
  hereby
    assume M is reduced;
    then M, the_reduction_of M-are_isomorphic by Th46;
    hence ex M being finite non empty Mealy-FSM over IAlph,OAlph st tfsm,
    the_reduction_of M-are_isomorphic;
  end;
  given MM being finite non empty Mealy-FSM over IAlph,OAlph such that
A1: M, the_reduction_of MM-are_isomorphic;
  set rMM = the_reduction_of MM;
  consider Tf being Function of the carrier of M, the carrier of rMM such that
A2: Tf is bijective and
  Tf.the InitS of M = the InitS of rMM and
A3: for q being State of M, s being Element of IAlph holds Tf.((the Tran
  of M).(q, s)) = (the Tran of rMM).(Tf.q, s) & (the OFun of M).(q,s)=(the OFun
  of rMM).(Tf.q, s) by A1;
  let qa, qb be State of M;
  assume qa <> qb;
  then Tf.qa <> Tf.qb by A2,FUNCT_2:19;
  then not Tf.qa, Tf.qb-are_equivalent by Th45;
  hence thesis by A3,Th44;
end;
