Theorem. uniqueness of $\mathbf{f}'(\mathbf{x})$ [br912]

Suppose $E$ and $\mathbf{f}$ are as in Definition 911, $\mathbf{x} \in E$, and $$ \lim_{ \mathbf{h} \to 0 } \frac{\lvert \mathbf{f}(\mathbf{x} + \mathbf{h}) - \mathbf{f}(\mathbf{x}) - A \mathbf{h} \rvert }{\lvert \mathbf{h} \rvert } = 0 $$ holds with $A = A_{1}$ and with $A = A_{2}$. Then $A_{1} = A_{2}$.