On differential inequalities of S.A. Chaplygin related to limit Cauchy problem for sets of ordinary differential equations of first order

We prove a theorem on differential inequalities related to limit Cauchy problem for the set of ordinary differential equations$$y' = f(x,y,z),$$z' = \varphi(x,y,z)$$with boundary conditions$$\lim\limits_{x \rightarrow \infty} y(x) = y(\infty) = y_0, \; \lim\limits_{x \rightarrow \infty} z(x) = z(\infty) = z_0$$