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
  for M1, M2 being connected reduced finite non empty Mealy-FSM over
  IAlph, OAlph holds M1, M2-are_isomorphic iff M1, M2-are_equivalent
proof
  let M1, M2 be connected reduced finite non empty Mealy-FSM over IAlph,
  OAlph;
  thus M1, M2-are_isomorphic implies M1, M2-are_equivalent by Th63;
A1: M2, the_reduction_of M2-are_isomorphic by Th46;
  assume M1, M2-are_equivalent;
  then
A2: the_reduction_of M1, the_reduction_of M2-are_isomorphic by Th65;
  M1, the_reduction_of M1-are_isomorphic by Th46;
  then M1, the_reduction_of M2-are_isomorphic by A2,Th42;
  hence thesis by A1,Th42;
end;
