Let $F:(\mathbb R\times \mathcal P (\mathbb R))\to \mathcal P (\mathbb R) \\ F((x,A))=\{y\in \mathbb R| \frac {x+y} 2\in A \}$
Define two different functions $g_k:\mathcal P (\mathbb R)\to (\mathbb R\times \mathcal P (\mathbb R))$ such that $F\circ g_k=i_{\mathcal P (\mathbb R)}, k=1,2$
This is equivalent to showing that $F$ is onto using composition of two different functions.
An easy example would be to take $F(0,\{1,2,3\})=\{2,4,6\}$ so for this example, this function could work $g(A)=\{x\in A| \frac x 2\}$.
But the $x$ in $\frac {x+y} 2$ really complicates things, I don't see a way to recover that $x$ when making the $g$'s so I can't revert the set back to have its original values..