Theorem 1

Let $a(x)$ and $b(x)$ be two functions belonging to a given Hardy field $\mathfrak{K}$ and $y(x)$ a function satisfying (1) on an interval $[ x_0, +\infty[$. When $r(u) = p(u)/q(u)$ runs through the set of rational fractions in $u$ with coefficients in $\mathfrak{K}$ such that $q(y)$ is not identically zero on a neighbourhood of $+\infty$ the set $\mathfrak{K}(y)$ of functions $r(y)$ forms a Hardy field.

Indeed, by lemma 2, there exists an interval $[ x_1, +\infty[$ on which $r(y)$ is defined, continuous, and does not change sign, from which it follows immediately that $\mathfrak{K}(y)/\mathbf{R}_\infty$ is a field; moreover, since

(where $r'(y) = (p'(y)q(y) - p(y)q'(y))/(q(y))^2$ is defined by hypothesis on a neighbourhood of $+\infty$), the derivative of every function in $\mathfrak{K}(y)$ belongs to $\mathfrak{K}(y)$, which proves that $\mathfrak{K}(y)$ satisfies the conditions of def. 1 of V, p. 247.

It is clear that $\mathfrak{K}(y)/\mathbf{R}_\infty$ is obtained by the algebraic adjunction to $\mathfrak{K}/\mathbf{R}_\infty$ of the class of $y$ modulo $\mathbf{R}_\infty$. Again one says that $\mathfrak{K}(y)$ is obtained by adjoining $y$ to $\mathfrak{K}$.