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;

theorem
  for tfsm being finite non empty Mealy-FSM over IAlph, OAlph st n+1 =
  card the carrier of tfsm holds ex k being Nat st k <= n & k
  -eq_states_partition tfsm is final
proof
  let tfsm be finite non empty Mealy-FSM over IAlph, OAlph;
  assume
A1: n+1 = card the carrier of tfsm;
  take n;
  thus n <= n;
  n-eq_states_partition tfsm = (n+1)-eq_states_partition tfsm by A1,Th35;
  hence thesis by Th37;
end;
