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 Th14:
  S is regular & S is calculating_type implies
  for s1, s2, q st q<>the InitS of S holds GEN(<*s1*>,q).2 = GEN(<*s2*>,q).2
proof
  assume that
A1: S is regular and
A2: S is calculating_type;
  let s1, s2, q;
  assume
A3: q<>the InitS of S;
  q is accessible by A1;
  then consider w such that
A4: the InitS of S,w-leads_to q;
  w <> {}
  by FSM_1:def 2,A3,A4;
  then consider x being Element of I, w9 being FinSequence of I such that
  w.1 = x and
A5: w = <*x*>^w9 by FINSEQ_3:102;
  set w1 = w^<*s1*>, w2 = w^<*s2*>;
  len <*x*> = 1 by FINSEQ_1:39;
  then len <*x*> + len w9 >= 1 by NAT_1:11;
  then len w >= 1 by A5,FINSEQ_1:22;
  then
A6: 1 in dom w by FINSEQ_3:25;
  then w.1 = w1.1 by FINSEQ_1:def 7;
  then
A7: w1.1 = w2.1 by A6,FINSEQ_1:def 7;
A8: len <*s1*> = 1 by FINSEQ_1:39;
  then
A9: len w1 = len w + 1 by FINSEQ_1:22;
A10: len <*s2*> = 1 by FINSEQ_1:39;
  then
A11: len w2 = len w + 1 by FINSEQ_1:22;
A12: len w1 = len w2 by A9,A10,FINSEQ_1:22;
  set p = Del(GEN(w, the InitS of S), len w + 1);
  set p1 = GEN(<*s1*>, q);
A13: GEN(w1, the InitS of S) = p^p1 by A4,FSM_1:8;
A14: len(GEN(w, the InitS of S)) = len w + 1 by FSM_1:def 2;
  then
A15: len p = len w by PRE_POLY:12;
A16: len p1 = len(<*s1*>) + 1 by FSM_1:def 2
    .= 1 + 1 by FINSEQ_1:39;
A17: len(GEN(w1, the InitS of S)) = len(p^p1) by A4,FSM_1:8
    .= len p + len p1 by FINSEQ_1:22
    .= len w + (1 + 1) by A14,A16,PRE_POLY:12
    .= (len w + 1) + 1
    .= len w1 + 1 by A8,FINSEQ_1:22;
A18: len p + 1 <= len w1 + 1 by A9,A15,NAT_1:11;
  len(p^p1) = len w1 + 1 by A4,A17,FSM_1:8;
  then len p + len p1 = len w1 + 1 by FINSEQ_1:22;
  then
A19: GEN(w1,the InitS of S).(len w1 + 1)=p1.((len w1 + 1) - len p)
  by A13,A18,FINSEQ_1:23
    .= p1.((len w1 + 1) - len w) by A14,PRE_POLY:12
    .= p1.((len w + 1 + 1) - len w) by A8,FINSEQ_1:22
    .= p1.2;
  set p2 = GEN(<*s2*>, q);
A20: GEN(w2, the InitS of S) = p^p2 by A4,FSM_1:8;
A21: len p2 = len(<*s2*>) + 1 by FSM_1:def 2
    .= 1 + 1 by FINSEQ_1:39;
A22: len(GEN(w2, the InitS of S)) = len(p^p2) by A4,FSM_1:8
    .= len p + len p2 by FINSEQ_1:22
    .= len w + (1 + 1) by A14,A21,PRE_POLY:12
    .= (len w + 1) + 1
    .= len w2 + 1 by A10,FINSEQ_1:22;
  len w + 1 <= len w2 + 1 by A11,NAT_1:11;
  then
A23: len p + 1 <= len w2 + 1 by A14,PRE_POLY:12;
  len(p^p2) = len w2 + 1 by A4,A22,FSM_1:8;
  then len w2 + 1 <= len p + len p2 by FINSEQ_1:22;
  then GEN(w2,the InitS of S).(len w2 + 1)=p2.((len w2 + 1) - len p)
  by A20,A23,FINSEQ_1:23
    .= p2.((len w2 + 1) - len w) by A14,PRE_POLY:12
    .= p2.((len w + 1 + 1) - len w) by A10,FINSEQ_1:22
    .= p2.2;
  hence thesis by A2,A7,A12,A19;
end;
