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 Th31:
  (k+1)-eq_states_partition tfsm <> k-eq_states_partition tfsm
  implies for i st i <= k holds (i+1)-eq_states_partition tfsm <> i
  -eq_states_partition tfsm
proof
  assume
A1: (k+1)-eq_states_partition tfsm <> k-eq_states_partition tfsm;
  let i be Nat such that
A2: i <= k;
A3: ex e being Nat st k+1 = i+e by A2,NAT_1:10,12;
  assume
A4: (i+1)-eq_states_partition tfsm = i-eq_states_partition tfsm;
  ex d being Nat st k = i+d by A2,NAT_1:10;
  then k-eq_states_partition tfsm = i-eq_states_partition tfsm by A4,Th29;
  hence contradiction by A1,A4,A3,Th29;
end;
