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;

theorem Th57:
  for crq11, crq12 being State of CRtfsm1 holds crq11 = q1u &
  crq12 = q2u & the carrier of CRtfsm1 misses the carrier of CRtfsm2 & tfsm =
CRtfsm1-Mealy_union CRtfsm2 & not crq11, crq12-are_equivalent implies not q1u,
  q2u-are_equivalent
proof
  let crq11, crq12 be State of CRtfsm1;
  set rtfsm1 = CRtfsm1, rtfsm2 = CRtfsm2, q1 = crq11, q2 = crq12;
  assume that
A1: q1 = q1u and
A2: q2 = q2u and
A3: (the carrier of rtfsm1) misses (the carrier of rtfsm2) & tfsm =
  rtfsm1 -Mealy_union rtfsm2;
  assume not q1,q2-are_equivalent;
  then consider w be FinSequence of IAlph such that
A4: (q1,w)-response <> (q2,w)-response;
  (q1u,w)-response = (q1,w)-response by A1,A3,Th53;
  then (q1u,w)-response <> (q2u,w)-response by A2,A3,A4,Th53;
  hence thesis;
end;
