# One-way function is not injective when it is in NP

Let us $$\Sigma = \{0,1\}$$ and $$f: \Sigma^* \rightarrow \Sigma^* \in FP$$ for which is valid that $$\exists k: \forall x \in \Sigma^* : \lvert x \rvert ^ {1/k} \leq \lvert f(x) \rvert \leq \lvert x \rvert ^ k$$. Thus, we can say that the function $$f$$ is one-way function.

We have language $$L = \{ w \; | \; \exists z \in \Sigma^*, w = f(z)\}$$. The question is, how to prove that $$f$$ is not injective if $$L \in NP \setminus UP$$, where $$UP$$ is the class of unambiguous TM.

It is clear, that if $$L \in NP \setminus UP$$ then exists NTM which decides this language and may exist at least one acceptable path for $$w \in L$$.