The notion of smooth biproximity space where $\delta_1,\delta_2$ are gradation proximities defined by Ghanim et al. [10]. In this paper, we show every smooth biproximity space $(X,\delta_1,\delta_2)$ induces a supra smooth proximity space $\delta_{12}$ finer than $\delta_1$ and $\delta_2$. We study the relationshipbetween $(X,\delta_{12})$ and the $FP^*$-separation axioms which had been introduced by Ramadan etal. [23]. Furthermore, we show for each smooth bitopological space which is $FP^*T_4$, the associated supra smooth topological space is a smooth supra proximal. The notion of $FP$-(resp. $FP^*$) proximity map are also introduced. In addition, we introduce the concept of $P$ smoothquasi-proximity spaces and prove that the associated smooth bitopological space $(X,\tau_\delta,\tau_{\delta^{-1}})$ satisfies $FP$-separation axioms in sense of Ramadan et al. [10]. تفاصيل المقالة