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;
reserve n, m, o, p for non zero Element of NAT,
  M for non empty Moore-SM_Final over I, O,
  q for State of M;

theorem
  for f being Function of {0, 1}, O holds
  Result(s, I-TwoStatesMooreSM(0,1,f)) = f.1
proof
  let f being Function of {0, 1}, O;
  set M = I-TwoStatesMooreSM(0,1,f);
  reconsider 01 = 1 as Element of {0, 1} by TARSKI:def 2;
A1: s leads_to_final_state_of M by Def6;
A2: the FinalS of M = {1} by Def7;
A3: the OFun of M = f by Def7;
  consider m being non zero Element of NAT such that
A4: for w st len w+1 >= m & w.1 = s holds
  GEN(w, the InitS of M).m in the FinalS of M and
A5: for w,j st j <= len w +1 & w.1 = s & j < m holds
  not GEN(w, the InitS of M).j in the FinalS of M by A1,Th18;
  now
    take m;
    let w;
    assume
A6: w.1 = s;
    hereby
      assume m <= len w + 1;
      then GEN(w, the InitS of M).m in the FinalS of M by A4,A6;
      hence f.1 = (the OFun of M).(GEN(w, the InitS of M).m) &
      GEN(w, the InitS of M).m in the FinalS of M by A2,A3,TARSKI:def 1;
    end;
    thus for n st n < m & n <= len w + 1 holds
    not GEN(w, the InitS of M).n in the FinalS of M by A5,A6;
  end;
  then f.01 is_result_of s,M;
  hence thesis by Def9;
end;
