the ring of real-continuous functions on a topoframe

a topoframe, denoted by $l_{ tau}$,  is a pair $(l, tau)$ consisting of a frame $l$ and a subframe $ tau $ all of whose elements are complementary elements in$l$. in this paper, we define and study the notions of a$tau $-real-continuous function on a frame $l$ and the set of realcontinuous functions $mathcal{r}l_tau $ as an $f$-ring.we show that $mathcal{r}l_{ tau}$is actually a generalization of the ring $c(x)$ of all real-valuedcontinuous functions on a completely regular hausdorff space $x$.in addition, we show that$mathcal{r}l_{ tau}$ is isomorphicto a sub-$f$-ring of $mathcal{r}tau .$let ${tau}$ be a topoframe on a frame $l$.the frame map $alphainmathcal{r}tau $ is called $l$-{it extendable} real continuous function if and only if for every$rin mathbb{r}$,$bigvee^{l}_{rin mathbb r} (alpha(-,r)veealpha(r,-))'=top.$finally, we prove that $mathcal{r}^{l}{tau}cong mathcal{r}l_{tau}$ as $f$-rings,where $mathcal{r}^{l}{tau}$ is the set all of $l$-extendable real continuous functions of $ mathcal{r}tau $.