Autonomous DNA computers have been attracting much attention because of their ability to integrate into living cells. Autonomous DNA computers can process information through DNA molecules and their molecular reactions. We have already proposed an idea of an autonomous molecular computer with high computational ability, which is now named Reverse-transcription-and-TRanscription-based Autonomous Computing System (RTRACS). In this study, we first report an experimental demonstration of a basic computation element of RTRACS and a mathematical modeling method for RTRACS. We focus on an AND gate, which produces an output RNA molecule only when two input RNA molecules exist, because it is one of the most basic computation elements in RTRACS. Experimental results demonstrated that the basic computation element worked as designed. In addition, its behaviors were analyzed using a mathematical model describing the molecular reactions of the RTRACS computation elements. A comparison between experiments and simulations confirmed the validity of the mathematical modeling method. This study will accelerate construction of various kinds of computation elements and computational circuits of RTRACS, and thus advance the research on autonomous DNA computers.