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 s, s1, s2, q st
  (the Tran of S).[the InitS of S,s1] <>
  (the Tran of S).[the InitS of S,s2] holds
  (the Tran of S).[q,s] <> the InitS of S
proof
  assume that
A1: S is regular and
A2: S is calculating_type;
  let s, s1, s2, q;
  assume
A3: (the Tran of S).[the InitS of S,s1] <> (the Tran of S).[the InitS of S,s2];
  q is accessible by A1;
  then consider w such that
A4: the InitS of S, w-leads_to q;
A5: GEN(w, the InitS of S).(len w + 1) = q by A4;
  assume
A6: (the Tran of S).[q,s]=the InitS of S;
  reconsider w1 = <*s,s1*>, w2 = <*s,s2*> as FinSequence of I;
A7: GEN(w1,q) = <* q, the InitS of S,
  (the Tran of S).[the InitS of S,s1] *> by A6,Th2;
A8: GEN(w2,q) = <* q, the InitS of S,
  (the Tran of S).[the InitS of S,s2] *> by A6,Th2;
A9: GEN(w1, q).3 = (the Tran of S).[the InitS of S,s1] by A7;
A10: GEN(w2, q).3 = (the Tran of S).[the InitS of S,s2] by A8;
A11: len w1 = 2 by FINSEQ_1:44;
A12: len w2 = 2 by FINSEQ_1:44;
A13: 3 <= len w1 +1 by A11;
A14: 3 <= len w2+1 by A12;
A15: GEN(w^w1, the InitS of S).(len w+3) = (the Tran of S).[the InitS of S,
  s1] by A4,A9,A13,FSM_1:7;
A16: GEN(w^w2, the InitS of S).(len w+3) = (the Tran of S).[the InitS of S,
  s2] by A4,A10,A14,FSM_1:7;
  per cases;
  suppose w = {};
    then
A17: len w = 0;
A18: GEN(w1, the InitS of S) = <* the InitS of S, (the Tran of S).[the InitS
    of S,s], (the Tran of S).[(the Tran of S).[the InitS of S,s], s1]*> by Th2;
A19: GEN(w2, the InitS of S) = <* the InitS of S, (the Tran of S).[the InitS
    of S,s], (the Tran of S).[(the Tran of S).[the InitS of S,s], s2]*> by Th2;
A20: w1.1 = s;
A21: w2.1 = s;
A22: 3 <= len w1 + 1 by A11;
A23: 3 <= len w2 + 1 by A12;
A24: GEN(w1, the InitS of S).3 =
    (the Tran of S).[(the Tran of S).[the InitS of S,s], s1]
    by A18
      .= (the Tran of S).[the InitS of S, s1]
    by A5,A6,A17,FSM_1:def 2;
    GEN(w2, the InitS of S).3 =
    (the Tran of S).[(the Tran of S).[the InitS of S,s], s2]
    by A19
      .= (the Tran of S).[the InitS of S, s2]
    by A5,A6,A17,FSM_1:def 2;
    hence contradiction by A2,A3,A20,A21,A22,A23,A24;
  end;
  suppose w <> {};
    then consider s9 being object, w9 being FinSequence such that
A25: w = <*s9*>^w9 and len w = len w9+1 by REWRITE1:5;
    dom <*s9*> = Seg 1 by FINSEQ_1:def 8;
    then
A26: 1 in dom <*s9*> by FINSEQ_1:1;
    then
A27: w.1 = <*s9*>.1 by A25,FINSEQ_1:def 7
      .= s9;
A28: dom <*s9*> c= dom w by A25,FINSEQ_1:26;
    then
A29: (w^w1).1 = s9 by A26,A27,FINSEQ_1:def 7;
A30: (w^w2).1 = s9 by A26,A27,A28,FINSEQ_1:def 7;
A31: len(w^w1)+1 = (len w+2)+1 by A11,FINSEQ_1:22;
    len(w^w2)+1 = (len w+2)+1 by A12,FINSEQ_1:22;
    hence contradiction by A2,A3,A15,A16,A29,A30,A31;
  end;
end;
