The proof of the uniqueness theorem given in this paper is based on the bound for a modulus of a differentiable function having many zeros in a small circle. Theorem. Let \(\mathcal V\) be an open connected subset of the complex plane, let a compact \(\mathcal K\) be contained in \(\mathcal V\) with some neighbourhood and a constructive function \(f\) be defined and differentiable in \(\mathcal V\). If there is \(u \in \mathcal K\) with \(|f(u)| > 0\) then \(z_1,\dots ,z_n \in \mathcal V\) a constant \(c\), and a constructive function \(g\) can be constructed such that \(f(z) = (z - z_1)\cdots (z - z_n ) g(z)\) for all \(z \in \mathcal V\), \(|g(z)| > c > 0\) for all \(z \in \mathcal K\). This theorem was proved by \textit{V. A. Lifshits} [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 20, 67--79 (1971; Zbl 0222.02030)] using Markov's principle. We prove it without using MP.