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
  k-eq_states_partition tfsm is final implies (k+1)-eq_states_EqR tfsm =
  k-eq_states_EqR tfsm
proof
  set S = the carrier of tfsm;
  set keq = k-eq_states_EqR tfsm;
  set k1eq = (k+1)-eq_states_EqR tfsm;
  set kpart = k-eq_states_partition tfsm;
  assume
A1: k-eq_states_partition tfsm is final;
  now
    let x be object;
    hereby
      assume
A2:   x in k1eq;
      then consider a, b be object such that
A3:   x=[a,b] and
A4:   a in S and
A5:   b in S by RELSET_1:2;
      reconsider b as Element of S by A5;
      reconsider a as Element of S by A4;
      (k+1)-equivalent a,b by A2,A3,Def12;
      then k-equivalent a,b by Th26;
      hence x in keq by A3,Def12;
    end;
    assume
A6: x in keq;
    then consider a, b be object such that
A7: x=[a,b] and
A8: a in S and
A9: b in S by RELSET_1:2;
    reconsider b as Element of S by A9;
    reconsider a as Element of S by A8;
    reconsider cl = Class(keq,b) as Element of kpart by EQREL_1:def 3;
A10: b in cl by EQREL_1:20;
    a in cl by A6,A7,EQREL_1:19;
    then a,b-are_equivalent by A1,A10;
    then (k+1)-equivalent a,b;
    hence x in k1eq by A7,Def12;
  end;
  hence thesis by TARSKI:2;
end;
