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;
reserve Rtfsm1, Rtfsm2 for reduced non empty Mealy-FSM over IAlph, OAlph;
reserve CRtfsm1, CRtfsm2 for connected reduced non empty Mealy-FSM over IAlph
  , OAlph,
  q1u, q2u for State of tfsm;
reserve CRtfsm1, CRtfsm2 for connected reduced finite non empty Mealy-FSM
  over IAlph, OAlph;

theorem Th60:
  tfsm = CRtfsm1-Mealy_union CRtfsm2 implies for Q being State of
  the_reduction_of tfsm holds not ex q1, q2 being Element of Q st q1 in the
  carrier of CRtfsm2 & q2 in the carrier of CRtfsm2 & q1 <> q2
proof
  set rtfsm1 = CRtfsm1;
  set rtfsm2 = CRtfsm2;
  set rtfsm = the_reduction_of tfsm;
  consider tfsm2 be finite non empty Mealy-FSM over IAlph, OAlph such that
A1: rtfsm2, the_reduction_of tfsm2-are_isomorphic by Th47;
  set tfsm2r = the_reduction_of tfsm2;
  consider Tf being Function of the carrier of rtfsm2, the carrier of tfsm2r
  such that
A2: Tf is bijective and
  Tf.the InitS of rtfsm2 = the InitS of tfsm2r and
A3: for q being State of rtfsm2, s being Element of IAlph holds Tf.((the
Tran of rtfsm2).(q, s)) = (the Tran of tfsm2r).(Tf.q, s) & (the OFun of rtfsm2)
  .(q,s) = (the OFun of tfsm2r).(Tf.q, s) by A1;
  assume
A4: tfsm = rtfsm1-Mealy_union rtfsm2;
  then
A5: the carrier of tfsm = (the carrier of rtfsm1) \/ (the carrier of rtfsm2
  ) by Def24;
  given Q be Element of the_reduction_of tfsm, q1, q2 being Element of Q such
  that
A6: q1 in the carrier of rtfsm2 and
A7: q2 in the carrier of rtfsm2 and
A8: q1 <> q2;
A9: dom Tf = the carrier of rtfsm2 by FUNCT_2:def 1;
  then
A10: Tf.q1 <> Tf.q2 by A6,A7,A8,A2,FUNCT_1:def 4;
  rng Tf=the carrier of tfsm2r by A2,FUNCT_2:def 3;
  then reconsider Tq1 = Tf.q1, Tq2 = Tf.q2 as Element of tfsm2r by A6,A7,A9,
FUNCT_1:def 3;
  reconsider q2 as Element of tfsm by A7,A5,XBOOLE_0:def 3;
  reconsider q29 = q2 as Element of rtfsm2 by A7;
  reconsider q1 as Element of tfsm by A6,A5,XBOOLE_0:def 3;
  reconsider q19 = q1 as Element of rtfsm2 by A6;
  not Tq1, Tq2 -are_equivalent by A10,Th45;
  then
A11: not q19,q29-are_equivalent by A3,Th44;
A12: the carrier of rtfsm = final_states_partition tfsm by Def18;
  then reconsider Q as Subset of tfsm by TARSKI:def 3;
A13: final_states_partition tfsm is final by Def15;
  q1,q2-are_equivalent by A13,A12;
  hence contradiction by A4,A11,Th58;
end;
