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 Th35:
  for tfsm being finite non empty Mealy-FSM over IAlph, OAlph st n
  +1 = card the carrier of tfsm holds (n+1)-eq_states_partition tfsm = n
  -eq_states_partition tfsm
proof
  let tfsm be finite non empty Mealy-FSM over IAlph, OAlph;
  assume
A1: n+1 = card the carrier of tfsm;
  defpred P[Nat] means $1 <= n+1 implies card ($1
  -eq_states_partition tfsm) > $1;
  assume
A2: (n+1)-eq_states_partition tfsm <> n-eq_states_partition tfsm;
A3: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume
A4: k <= n+1 implies card (k-eq_states_partition tfsm) > k;
    assume
A5: k+1 <= n+1;
    then k <= n by XREAL_1:6;
    then
A6: (k+1)-eq_states_partition tfsm<>k-eq_states_partition tfsm by A2,Th31;
    k+1 <= card (k-eq_states_partition tfsm) by A4,A5,NAT_1:13;
    hence thesis by A6,Th32,XXREAL_0:2;
  end;
A7: P[0];
  for k being Nat holds P[k] from NAT_1:sch 2(A7, A3);
  then card ((n+1)-eq_states_partition tfsm) > n+1;
  hence contradiction by A1,FINSEQ_4:88;
end;
