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 Th28:
  (k+1)-eq_states_partition tfsm is_finer_than k -eq_states_partition tfsm
proof
  set K = k-eq_states_partition tfsm;
  set K1 = (k+1)-eq_states_partition tfsm;
  set S = the carrier of tfsm;
  let X be set;
  assume
A1: X in K1;
  then reconsider X9 = X as Subset of S;
  consider q being Element of S such that
A2: q in X by A1,FINSEQ_4:87;
  reconsider Y = (proj K).q as Element of K;
  take Y;
  thus Y in K;
  let x be object;
  assume
A3: x in X;
  then x in X9;
  then reconsider x9=x as Element of S;
  reconsider X9 as Element of Class ((k+1)-eq_states_EqR tfsm) by A1;
  consider Q being object such that
  Q in S and
A4: X9 = Class ((k+1)-eq_states_EqR tfsm, Q) by EQREL_1:def 3;
  [x9, Q] in (k+1)-eq_states_EqR tfsm by A3,A4,EQREL_1:19;
  then
A5: [Q, x9] in (k+1)-eq_states_EqR tfsm by EQREL_1:6;
  [q, Q] in (k+1)-eq_states_EqR tfsm by A2,A4,EQREL_1:19;
  then [q, x9] in (k+1)-eq_states_EqR tfsm by A5,EQREL_1:7;
  then (k+1)-equivalent q, x9 by Def12;
  then k-equivalent q, x9 by Th26;
  then [q, x9] in k-eq_states_EqR tfsm by Def12;
  then
A6: [x9, q] in k-eq_states_EqR tfsm by EQREL_1:6;
  reconsider Y9 = Y as Element of Class (k-eq_states_EqR tfsm);
  consider Q being object such that
A7: Q in S and
A8: Y9 = Class (k-eq_states_EqR tfsm, Q) by EQREL_1:def 3;
  reconsider Q as Element of S by A7;
  q in Y by EQREL_1:def 9;
  then Class(k-eq_states_EqR tfsm,Q)=Class(k-eq_states_EqR tfsm,q)by A8,
EQREL_1:23;
  hence thesis by A6,A8,EQREL_1:19;
end;
