Several things: (i) The set $\mathbb{R}$ is denoted by $\mathbb{R}_0$ (unnecessary). (ii) On $\mathbb{R}_0\times\mathbb{R}_0$ the following relation is defined $(x_1,x_2)=(y_1,y_2)\Leftrightarrow x_1y_2=x_1y_1$ (very bad notation, it should be for example $(x_1,x_2)\sim(y_1,y_2)\Leftrightarrow x_1y_2=x_1y_1$ . (iii) But the worst thing is that $\sim$ is not an equivalence relation, so the given operations have no sense. Could you exactly transcribe the formulation of the problem?. Perhaps you forgot to say that $\mathbb{R}_0=\mathbb{R}-\{0\}$ .