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 Th17:
  for i,f being set, o being Function of {i,f}, O
  for j st 1 < j & j <= len w+1 holds
  GEN(w, the InitS of I-TwoStatesMooreSM(i,f,o)).j = f
proof
  let i,f being set, o being Function of {i,f},O;
  let j;
  assume
A1: 1 < j;
  set M = I-TwoStatesMooreSM(i,f,o);
A2: the carrier of M = {i, f} by Def7;
A3: the Tran of M = [:{i, f}, I:] --> f by Def7;
  defpred P[Nat] means $1 <= len w + 1 implies
  GEN(w, the InitS of I-TwoStatesMooreSM(i,f,o)).$1 = f;
A4: P[2]
  proof
    assume 2 <= len w + 1;
    then 1+1 <= len w+1;
    then 1 < len w+1 by NAT_1:13;
    then 1 <= len w by NAT_1:13;
    then ex WI being Element of I, QI, QI1 being State of M st ( WI
= w.1)&( QI = GEN(w, the InitS of M).1)&( QI1 = GEN(w, the InitS of M).(1+1))&(
    WI-succ_of QI = QI1) by FSM_1:def 2;
    hence thesis by A2,A3,FUNCOP_1:7;
  end;
A5: for k be non trivial Nat st P[k] holds P[k+1]
  proof
    let k be non trivial Nat;
    assume that k <= len w + 1 implies GEN(w, the InitS of M).k = f and
A6: k+1 <= len w + 1;
A7: 1 <= k by NAT_2:19;
    k <= len w by A6,XREAL_1:6;
    then ex WI being Element of I, QI, QI1 being State of M st ( WI
= w.k)&( QI = GEN(w, the InitS of M).k)&( QI1 = GEN(w, the InitS of M).(k+1))&(
    WI-succ_of QI = QI1) by A7,FSM_1:def 2;
    hence thesis by A2,A3,FUNCOP_1:7;
  end;
A8: j is non trivial Nat by A1,NAT_2:def 1;
  for k be non trivial Nat holds P[k] from NAT_2:sch 2(A4,A5);
  hence thesis by A8;
end;
