diff --git a/thue-morse.v b/thue-morse.v
index f190a16..a1a68f6 100644
--- a/thue-morse.v
+++ b/thue-morse.v
@@ -1378,9 +1378,7 @@ Proof.
   rewrite tm_size_power2. apply lt_split_bits.
   assumption. assumption. assumption.
   replace (nth_error (tm_step m) (2^j)) with (nth_error (tm_step (S j)) (2^j)).
-  rewrite tm_build. rewrite nth_error_app2. rewrite tm_size_power2.
-  rewrite Nat.sub_diag. rewrite tm_step_head_1. simpl. reflexivity.
-  rewrite tm_size_power2. apply Nat.le_refl.
+  rewrite tm_step_single_bit_index. reflexivity.
   apply tm_step_stable. rewrite <- Nat.mul_1_l at 1.
   rewrite Nat.pow_succ_r. rewrite <- Nat.mul_lt_mono_pos_r.
   apply Nat.lt_1_2.