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

theorem Th11:
  i <= j & Following (Computation s).i = (Computation s).i implies
  (Computation s).j = (Computation s).i
proof
  assume that
A1: i <= j and
A2: Following (Computation s).i = (Computation s).i;
  consider k be Nat such that
A3: j = i + k by A1,NAT_1:10;
  defpred X[Nat] means
   (Computation s).(i+$1) = (Computation s).i;
A4: for k st X[k] holds X[k+1]
  proof
    let k;
    assume
A5: (Computation s).(i+k) = (Computation s).i;
    thus (Computation s).(i+(k+1)) = (Computation s).(i+k+1)
      .= (Computation s).i by A2,A5,Def7;
  end;
A6: X[0];
A7: for k holds X[k] from NAT_1:sch 2(A6,A4);
  thus thesis by A3,A7;
end;
