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
  for S st (for s1,s2,q st q<>the InitS of S holds
  (the Tran of S).[q,s1] = (the Tran of S).[q,s2]) & for s,q1 holds
  (the Tran of S).[q1,s] <> the InitS of S holds S is calculating_type
proof
  let S;
  assume that
A1: for s1,s2,q st q<>the InitS of S holds (the Tran of S).[q,s1] = (the
  Tran of S).[q,s2] and
A2: for s,q1 holds (the Tran of S).[q1,s] <> the InitS of S;
A3: for j st j >= 2 holds for w1 st j <= len(w1)+1 holds
  GEN(w1, the InitS of S).j <> the InitS of S
  proof
    let j;
    assume j >= 2;
    then j <> 1;
    then
A4: j is non trivial by NAT_2:def 1;
    defpred P[Nat] means for w1 st $1 <= len w1 + 1 holds
    GEN(w1, the InitS of S).$1 <> the InitS of S;
A5: P[2]
    proof
      let w1;
      assume 2 <= len(w1) + 1;
      then 1 + 1 <= len(w1) + 1;
      then 1 <= len(w1) by XREAL_1:6;
      then ex WI being Element of I, QI, QI1 being State of S st ( WI =
w1.1)&( QI = GEN(w1, the InitS of S).1)&( QI1 = GEN(w1, the InitS of S).(1+1))&
      ( WI-succ_of QI = QI1) by FSM_1:def 2;
      hence thesis by A2;
    end;
A6: for h being non trivial Nat st P[h] holds P[h+1]
    proof
      let h be non trivial Nat;
      assume for w1 st h <= len(w1)+1 holds
      GEN(w1, the InitS of S).h <> the InitS of S;
      let w1;
      assume
A7:   h+1 <= len(w1)+1;
A8:   1 <= h by NAT_1:14;
      h <= len(w1) by A7,XREAL_1:6;
      then ex WI being Element of I, QI, QI1 being State of S st ( WI =
w1.h)&( QI = GEN(w1, the InitS of S).h)&( QI1 = GEN(w1, the InitS of S).(h+1))&
      ( WI-succ_of QI = QI1) by A8,FSM_1:def 2;
      hence thesis by A2;
    end;
    for h being non trivial Nat holds P[h] from NAT_2:sch 2(A5,A6);
    hence thesis by A4;
  end;
  for j holds for w1,w2 st w1.1 = w2.1 & j <= len(w1)+1 & j <= len(w2)+1 holds
  GEN(w1, the InitS of S).j = GEN(w2, the InitS of S).j
  proof
    let j;
    let w1,w2;
    assume that
A9: w1.1 = w2.1 and
A10: j <= len(w1)+1 and
A11: j <= len(w2)+1;
    defpred P[Nat] means for w1,w2 st w1.1 = w2.1 &
    $1 <= len(w1) + 1 & $1 <= len(w2) + 1
    holds GEN(w1, the InitS of S).$1 = GEN(w2, the InitS of S).$1;
A12: P[1]
    proof
      let w1,w2;
      GEN(w1, the InitS of S).1 = the InitS of S by FSM_1:def 2;
      hence thesis by FSM_1:def 2;
    end;
A13: for h being non zero Nat st P[h] holds P[h+1]
    proof
      let h be non zero Nat;
      assume
A14:  for w1,w2 st w1.1 = w2.1 & h <= len(w1)+1 & h <= len(w2)+1 holds
      GEN(w1, the InitS of S).h = GEN(w2, the InitS of S).h;
      let w1,w2;
      assume that
A15:  w1.1 = w2.1 and
A16:  h+1 <= len(w1)+1 and
A17:  h+1 <= len(w2)+1;
A18:  h <= len(w1) by A16,XREAL_1:6;
A19:  h <= len(w1)+1 by A16,NAT_1:13;
A20:  1 <= h by NAT_1:14;
      then consider WI being Element of I, QI, QI1 being State of S such that
A21:  WI = w1.h and
A22:  QI = GEN(w1, the InitS of S).h and
A23:  QI1 = GEN(w1, the InitS of S).(h+1) and
A24:  WI-succ_of QI = QI1 by A18,FSM_1:def 2;
A25:  h <= len(w2) by A17,XREAL_1:6;
A26:  h <= len(w2)+1 by A17,NAT_1:13;
      1 <= h by NAT_1:14;
      then consider WI2 being Element of I,
      QI2, QI12 being State of S such that
A27:  WI2 = w2.h and
A28:  QI2 = GEN(w2, the InitS of S).h and
A29:  QI12 = GEN(w2, the InitS of S).(h+1) and
A30:  WI2-succ_of QI2 = QI12 by A25,FSM_1:def 2;
A31:  QI = QI2 by A14,A15,A19,A22,A26,A28;
A32:  h in NAT by ORDINAL1:def 12;
      h = 1 or h > 1 by A20,XXREAL_0:1;
      then h = 1 or h >= 1 + 1 by NAT_1:13;
      hence thesis by A1,A3,A15,A19,A21,A22,A23,A24,A27,A29,A30,A31,A32;
    end;
    for j being non zero Nat holds P[j] from NAT_1:sch 10(A12,A13);
    hence thesis by A9,A10,A11;
  end;
  hence thesis;
end;
