reserve n,i,j,k for Nat;
reserve T for TuringStr,
  s for All-State of T;

theorem Th13:
  for T being TuringStr,s be All-State of T st s is Accept-Halt
   ex k being Nat st
  ((Computation s).k)`1_3 = the AcceptS of T &
  Result s = (Computation s).k & for i be Nat st i < k holds ((
  Computation s).i)`1_3 <> the AcceptS of T
proof
  let T be TuringStr,s be All-State of T;
  defpred P[Nat] means ((Computation s).$1)`1_3 = the AcceptS of T;
  assume
A1: s is Accept-Halt;
  then ex k st ((Computation s).k)`1_3 = the AcceptS of T;
  then
A2: ex k be Nat st P[k];
  consider k being Nat such that
A3: P[k] & for n be Nat st P[n] holds k <= n from NAT_1:sch 5(A2);
  reconsider k as Element of NAT by ORDINAL1:def 12;
  take k;
  thus P[k] by A3;
  thus Result s=(Computation s).k by A1,A3,Def9;
  thus thesis by A3;
end;
