countable composition closedness and integer-valued continuous functions in pointfree topology

‎for any archimedean$f$-ring $a$ with unit in whichbreak$awedge‎‎(1-a)leq 0$ for all $ain a$‎, ‎the following are shown to be‎‎equivalent‎:‎‎1‎. ‎$a$ is isomorphic to the $l$-ring ${mathfrak z}l$ of all‎‎integer-valued continuous functions on some frame $l$‎. 2‎. ‎$a$ is a homomorphic image of the $l$-ring $c_{bbb z}(x)$‎‎of all integer-valued continuous functions‎, ‎in the usual sense‎,‎on some topological space $x$‎.3‎. ‎for any family $(a_n)_{nin omega}$ in $a$ there exists an‎‎$l$-ring homomorphism break$varphi‎ :‎c_{bbb z}(bbb‎‎z^omega)rightarrow a$ such that $varphi(p_n)=a_n$ for the‎‎product projections break$p_n:{bbb z^omega}rightarrow bbb z$‎.‎this provides an integer-valued counterpart to a familiar result‎‎concerning real-valued continuous functions‎.