In this paper, the authors prove (Theorem~1.1): Let \(f\), \(g\) be distinct nonconstant meromorphic functions and \(n\geq 10\) be an integer. Put \(h=f/g\). If \(E(\infty,f)=E(\infty,g)\) and \(E_{(3)}\bigl(1,f^n(f-1)f'\bigr)=E_{(3)}\bigl(1,g^n(g-1)g'\bigr)\), then \(f=R_n(1/h)\) and \(g=R_n(h)\) for the rational function \(R_n(w)=P_{n+1}(w)/P_{n+2}(w)\) with \(P_k(w)=(w^{k}-1)/k\). As its corollary, if one restricts both \(f\) and \(g\) to entire functions, the uniqueness \(f\equiv g\) is obtained from the \textit{set} equation \(E_{(3)}\bigl(1,f^n(f-1)f'\bigr)=E_{(3)}\bigl(1,g^n(g-1)g'\bigr)\) when \(n\geq 7\). Here \(E_{(k)}(a,\varphi)\) (\(k\in\mathbb{N}\cup\{\infty\}\), \(a\in\mathbb{C}\cup\{\infty\}\)) is the sequence of the \(a\)-points of a nonconstant meromorphic function~\(\varphi\) in~\(\mathbb{C}\) with multiplicity \(\leq k\), each points being repeated as many times as its multiplicity. Especially, \(E(a,\varphi)=E_{(\infty)}(a,\varphi)\). These results give extensions of some earlier results by \textit{M. L. Fang} and \textit{W. Hong} [Indian J. Pure Appl. Math. 32, No. 9, 1343--1348 (2001; Zbl 1005.30023)], \textit{H. L. Qiu} and \textit{Fang} [Bull. Korean Math. Soc. 41, No.1, 109-116 (2004; Zbl 1134.30325)], \textit{W. C. Lin} and \textit{H. X. Yi} in [Indian J. Pure Appl. Math. 35, No. 2, 121--132 (2004; Zbl 1056.30031)], and others. Related results are given by \textit{I. Lahiri} and \textit{N. Mandal} in [Int. J. Math. Math. Sci. 2005, No. 12, 1933--1942 (2005; Zbl 1084.30029)], too. The corollary itself seems to be proved also by \textit{I. Lahiri} and \textit{P. Sahoo} in [Georgian Math. J. 12, No. 1, 131--138 (2005; Zbl 1073.30022)]. One needs an additional assumption in order to exclude the case when \(f=R_n(1/h)\) and \(g=R_n(h)\) and then to conclude the uniqueness of transcendental meromorphic~\(f\) and~\(g\) under consideration. Some of the papers above limit the frequency of their poles with \(n\), such as \(\Theta(\infty,\varphi)>2/(n+1)\), since the \textit{deficiency} attains \(2/(n+1)\) for \(\varphi=R_n(h)\) with a suitable~\(h\). A proof of such results requires a \textit{variant} of the so-called Nevanlinna-Cartan Second Main Theorem for \textit{distinct} meromorphic functions~\(F\) and~\(G\) sharing \(1\)-points with \(FG\not\equiv 1\), as Lemma~2.2 in the paper under review. Then they apply their estimate to \(F=\Pi(f)\) and \(G=\Pi(g)\) with \(\Pi(\varphi):=\varphi^n(\varphi-1)\varphi'\). (For example, Lahiri and Mandal show that \(\Pi(f)\Pi(g)\not\equiv 1\) for any nonconstant meromorphic functions~\(f\) and~\(g\) when \(n\geq 5\).) This differential polynomial \(\Pi\bigl(\varphi(z)\bigr)\) is the derivative of \(P_{n+2}\bigl(\varphi(z)\bigr)-P_{n+1}\bigl(\varphi(z)\bigr)\) and the exceptional pair in Theorem~1.1, \(R_n(1/h)\) and \(R_n(h)\), is obtained when \(F\equiv G\) holds with \(h\not\equiv 1\). In fact, they prove that \(P_{n+2}(f)-P_{n+1}(f)\equiv P_{n+2}(g)-P_{n+1}(g)\) still holds, which implies directly the desired expression with \(h=f/g\). One could think it natural that the methods and discussions are overlapping those in many papers relating to the famous problems on \textit{uniqueness polynomials} posed by \textit{F. Gross} in [Complex Anal., Proc. Conf., Lexington 1976, Lect. Notes Math. 599, 51--67 (1977; Zbl 0357.30007)] and on differential polynomials by [\textit{W. K. Hayman}, Research problems in function theory. London: University of London. The Athlone Press (1967; Zbl 0158.06301)], respectively.