Uniqueness of Continuously Differentiable Solution to Initial Value Problem

Theorem
Let $D \subseteq \R^{2}$ be a region containing $\left({a, b}\right)$.

Let $f: D \to \R$ be real-valued mapping such that:
 * $f$ is continuously differentiable on $D$, and
 * $\dfrac{\partial f}{\partial x}$ is continuously differentiable on $D$

Consider the initial value problem:
 * $\dfrac {\mathrm d x} {\mathrm d t} = f \left({t, x}\right)$
 * $x \left({ a }\right) = b$

Suppose the initial value problem above has a solution $\phi$ for all $x$ in some  interval $J$ containing $a$.

Then the solution $\phi$ is unique on $J$.