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

theorem Th10:
  (Computation s).(i+k) = (Computation (Computation s).i).k
proof
  defpred X[Nat] means
  (Computation s).(i+$1) = (Computation (Computation s).i).$1;
A1: for k st X[k] holds X[k+1]
  proof
    let k;
    assume
A2: (Computation s).(i+k) = (Computation (Computation s).i).k;
    thus (Computation s).(i+(k+1)) = (Computation s).(i+k+1)
      .= Following (Computation s).(i+k) by Def7
      .= (Computation (Computation s).i).(k+1) by A2,Def7;
  end;
A3: X[0] by Def7;
  for k holds X[k] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
