Mapping prefer-opposite to prefer-one de Bruijn sequences

We present a mapping of the binary prefer-opposite de Bruijn sequence of order n onto the binary prefer-one de Bruijn sequence of order n- 1. The mapping is based on the differentiation operator D(⟨ b₁, … , b_l⟩) = ⟨ b₂- b₁, b₃- b₂, … , b_l- b_{l - 1}⟩ where bit subtraction is modulo two. We show that if we take the prefer-opposite sequence ⟨b1,b2,…,b2n⟩, apply D to get the sequence ⟨b^1,…,b^2n-1⟩ and drop all the bits b^ _i such that ⟨ b^ _i, … , b^ _{i + n - 1}⟩ is a substring of ⟨ b^ ₁, … , b^ _{i + n - 2}⟩ , we get the prefer-one de Bruijn sequence of order n- 1.