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 Th15:
  tfsm1, tfsm2-are_equivalent & tfsm2, tfsm3-are_equivalent
  implies tfsm1, tfsm3-are_equivalent
proof
  assume that
A1: tfsm1, tfsm2-are_equivalent and
A2: tfsm2, tfsm3-are_equivalent;
  let w1 be FinSequence of IAlph;
  set IS3 = the InitS of tfsm3;
  set IS1 = the InitS of tfsm1, IS2 = the InitS of tfsm2;
  thus (IS1, w1)-response = (IS2, w1)-response by A1
    .= (IS3, w1)-response by A2;
end;
