Introduce $\hat A$ and $\hat B$ as in \[lem:duality\]. Clearly $G$ is also an ${\bm{N}}$-minimal approximant basis of $\hat B$ of order $d$. Likewise, $\hat A$ and $A$ have the same minimal approximant bases for given order and shift.

Assume, without loss of generality, that we have scaled $G$ such that $\det G$ is monic. Since $\hat A{^\top}$ is also an approximant basis for $\hat B$ of order $d$, then $\det G = \det \hat A{^\top}= x^d$. By definition $G\hat B = x^d R$ for some matrix $R \in
  {{\mathsf{K}}}[x]^{(n+1)\times(n+1)}$. That means $$\begin{aligned}
    x^{2d}((G\hat B){^\top}))^{{-1}}&= x^{2d}((x^d R){^\top})^{{-1}}\ , & \textrm{so} \\ 
    (x^d(G{^\top})^{{-1}})(x^d(\hat B{^\top})^{{-1}}) &= x^{d}(R{^\top})^{{-1}}\ , & \textrm{that is} \\ 
    {{\textnormal{adj}}}(G{^\top}) \hat A &= x^d (R{^\top})^{{-1}}\ .
  \end{aligned}$$ Now $\det R = 1$ since $(x^d)^{n+1} \det R = \det(G\hat B) =
x^{d+nd}$, so $(R{^\top})^{{-1}}= {{\textnormal{adj}}}(R{^\top}) \in {{\mathsf{K}}}[x]^{(n+1)
\times (n+1)}$. Therefore ${{\textnormal{adj}}}(G{^\top})$ is an approximant basis of $\hat A$ of order $d$. The theorem now follows from \[lem:adjointRowReduced\] by noting that $G$ is ${\bm{N}}$-row reduced.

We apply \[thm:dualityMinbasis\] to the problem of \[ex:simpade\] with shifts ${\bm{N}} = (5, 3, 4, 5)$. We have $$\begin{aligned}
    A &= 
      \left[\begin{array}{rrr}
      x^{4} + x^{2} + 1 & x^{4} + 1 & x^{4} + x^{3} + 