The theorem reads:
Let $T:\mathbb R^n\longrightarrow \mathbb R^n$ be an isometry. Then $T$ is an isomorphism.
Let $\mathbf u_1,\mathbf u_2,\ldots,\mathbf u_n$ be an orthonormal basis for $\mathbb R^n$. Then $T\mathbf u_1,T\mathbf u_2,\ldots,T\mathbf u_n$ is an orthonormal basis for $\mathbb R^n$. Then image $T=\mathbb R^n$.
I see that it was proved that $T$ must be onto, but I don't see why $T$ is 1-1. One would have to prove that for each $\mathbf x,\mathbf y \in \mathbb R^n$, $T\mathbf x = T \mathbf y\Longrightarrow \mathbf x = \mathbf y$. I appreciate your help in pointing out a hint to prove the last part.