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 Th23:
  for k be Nat st k-equivalent qa, qb & k-equivalent qb, qc holds
  k-equivalent qa, qc
proof
  let k be Nat;
  assume that
A1: k-equivalent qa, qb and
A2: k-equivalent qb, qc;
  thus k-equivalent qa, qc
  proof
    let w be FinSequence of IAlph;
    assume
A3: len w <= k;
    then (qa, w)-response = (qb, w)-response by A1;
    hence thesis by A2,A3;
  end;
end;
