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;
