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 Th32:
  for tfsm being finite non empty Mealy-FSM over IAlph, OAlph
  holds k-eq_states_partition tfsm = (k+1)-eq_states_partition tfsm or card (k
  -eq_states_partition tfsm) < card ((k+1)-eq_states_partition tfsm)
proof
  let tfsm be finite non empty Mealy-FSM over IAlph, OAlph;
  set kp = k-eq_states_partition tfsm;
  set k1p = (k+1)-eq_states_partition tfsm;
  card kp <= card k1p by Th28,FINSEQ_4:89;
  then
A1: card kp = card k1p or card kp < card k1p by XXREAL_0:1;
  assume kp <> k1p;
  hence thesis by A1,Th28,FINSEQ_4:91;
end;
