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 Th59:
  the carrier of CRtfsm1 misses the carrier of CRtfsm2 & 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 CRtfsm1 & q2 in
  the carrier of CRtfsm1 & q1 <> q2
proof
  set rtfsm1 = CRtfsm1;
  set rtfsm2 = CRtfsm2;
  assume that
A1: (the carrier of rtfsm1) misses (the carrier of rtfsm2) and
A2: tfsm = rtfsm1-Mealy_union rtfsm2;
  given Q be Element of the_reduction_of tfsm, q1, q2 being Element of Q such
  that
A3: q1 in the carrier of rtfsm1 & q2 in the carrier of rtfsm1 and
A4: q1 <> q2;
  consider tfsm1 be finite non empty Mealy-FSM over IAlph, OAlph such that
A5: rtfsm1, the_reduction_of tfsm1-are_isomorphic by Th47;
  set tfsm1r = the_reduction_of tfsm1;
  consider Tf being Function of the carrier of rtfsm1, the carrier of tfsm1r
  such that
A6: Tf is bijective and
  Tf.the InitS of rtfsm1 = the InitS of tfsm1r and
A7: for q being State of rtfsm1, s being Element of IAlph holds Tf.((the
Tran of rtfsm1).(q, s)) = (the Tran of tfsm1r).(Tf.q, s) & (the OFun of rtfsm1)
  .(q,s) = (the OFun of tfsm1r).(Tf.q, s) by A5;
A8: dom Tf = the carrier of rtfsm1 by FUNCT_2:def 1;
  then
A9: Tf.q1 <> Tf.q2 by A3,A4,A6,FUNCT_1:def 4;
  rng Tf=the carrier of tfsm1r by A6,FUNCT_2:def 3;
  then reconsider Tq1 = Tf.q1, Tq2 = Tf.q2 as Element of tfsm1r by A3,A8,
FUNCT_1:def 3;
  the carrier of tfsm = (the carrier of rtfsm1) \/ (the carrier of rtfsm2
  ) by A2,Def24;
  then reconsider q1, q2 as Element of tfsm by A3,XBOOLE_0:def 3;
  reconsider q19 = q1, q29 = q2 as Element of rtfsm1 by A3;
  not Tq1, Tq2 -are_equivalent by A9,Th45;
  then
A10: not q19,q29-are_equivalent by A7,Th44;
  set rtfsm = the_reduction_of tfsm;
A11: final_states_partition tfsm is final by Def15;
A12: the carrier of rtfsm = final_states_partition tfsm by Def18;
  then reconsider Q as Subset of tfsm by TARSKI:def 3;
  q1,q2-are_equivalent by A11,A12;
  hence contradiction by A1,A2,A10,Th57;
end;
