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
  0-eq_states_partition tfsm = { the carrier of tfsm }
proof
  set S = the carrier of tfsm;
  set SS = { the carrier of tfsm };
  set 0p = 0-eq_states_partition tfsm;
  set 0e = 0-eq_states_EqR tfsm;
  now
    set y = the Element of S;
    let x be object;
    hereby
      assume
A1:   x in 0p;
      then reconsider x9 = x as Subset of S;
      consider y being object such that
A2:   y in S and
A3:   x9 = Class (0e, y) by A1,EQREL_1:def 3;
      reconsider y as Element of S by A2;
      Class (0e, y) = S by Th33;
      hence x in SS by A3,TARSKI:def 1;
    end;
    assume x in SS;
    then
A4: x = S by TARSKI:def 1;
    Class (0e, y) in Class 0e by EQREL_1:def 3;
    hence x in 0p by A4,Th33;
  end;
  hence thesis by TARSKI:2;
end;
