A Note about Stabilization in $A_\R(\D)$

It is shown that for $A_\R(\D)$ functions $f_1$ and $f_2$ with $$\inf_{z\in\bar{\D}}(\abs{f_1(z)}+\abs{f_2(z)})\geq\delta>0$$ and $f_1$ being positive on real zeros of $f_2$ then there exists $A_\R(\D)$ functions $g_2$ and $g_1,g_1^{-1}$ with and $$g_1f_1+g_2f_2=1\quad\forall z\in\bar{\D}.$$ This result is connected to the computation of the stable rank of the algebra $A_\R(\D)$ and to Control Theory.