Could you give me a hint how to solve this problem?
Let $ D:= \left\{E \subset \mathbb{R} \ | \ 0< \mathrm{card}(E)< + \infty \right\} $. $\phi\colon D\to\mathbb R$ defined by $\phi(E)\sum_{x \in E}x$
Check if $\phi$ is injective or surjective.
Well, I think that it can't be injective because $\mathrm{card}(\mathcal P(\mathbb{R}))>\mathrm{card}(\mathbb{R})$ , so by the pigeonhole principle, there must to at least two subsets with the same sum of elements.
Hint: The fact that $\phi$ is not injective follows from the fact that $\{0,x\}$ and $\{x\}$ have the same sum. Surjectivity follows from a similar argument as well.
Observation: $D$ has the same cardinality as $\mathbb R$.