reserve x,y for Real,
  i, j for non zero Element of NAT,
  I, O for non empty set,
  s,s1,s2,s3 for Element of I,
  w, w1, w2 for FinSequence of I,
  t for Element of O,
  S for non empty FSM over I,
  q, q1 for State of S;

theorem
  S is regular & S is calculating_type implies
  for s1, s2, q st q<>the InitS of S holds
  (the Tran of S).[q,s1] = (the Tran of S).[q,s2]
proof
  assume that
A1: S is regular and
A2: S is calculating_type;
  let s1, s2, q;
  assume
A3: q<>the InitS of S;
  set w1=<*s1*>;
  set w2=<*s2*>;
A4: len w1 = 0+1 by FINSEQ_1:39;
  reconsider j = len w1 as non zero Element of NAT;
  consider WI being Element of I, QI, QI1 being State of S such that
A5: WI = w1.j and
A6: QI = GEN(w1, q).j and
A7: QI1 = GEN(w1, q).(j+1) and
A8: WI-succ_of QI = QI1 by A4,FSM_1:def 2;
  WI = s1 by A4,A5;
  then
A9: QI1 = s1-succ_of q by A4,A6,A8,FSM_1:def 2;
A10: len w2 = 0+1 by FINSEQ_1:39;
  reconsider h = len w2 as non zero Element of NAT;
  consider WI2 being Element of I, QI2, QI12 being State of S such that
A11: WI2 = w2.h and
A12: QI2 = GEN(w2, q).h and
A13: QI12 = GEN(w2, q).(h+1) and
A14: WI2-succ_of QI2 = QI12 by A10,FSM_1:def 2;
A15: QI2 = q by A10,A12,FSM_1:def 2;
  WI2 = s2 by A10,A11;
  hence thesis by A1,A2,A3,A4,A7,A9,A10,A13,A14,A15,Th14;
end;
