Dual version of a generalization of Erdös-Beck theorem

Question

In the Paper of Do she states that her theorem 1.6 also holds if one embeds $\mathbb{R}^d$ into $\mathbb{R}\mathbb{P}^d$, the theorem is:

For any $0 < \beta < 1$ there is some constand $c(\beta)$ depending on $d$ and $\beta$ such that for any set $S$ of $n$ points in $\mathbb{R}^d$, either

1. there exists a collection of flats $\{F_1,...,F_k\}$ whose union contains $\beta n$ points of $S$ and $\sum_{i=1}^d \dim F_i<d$

or

2. the number of hyperplanes spanned by the points of $S$ exceeds $c(\beta) \cdot n^d$.

I want to dualize this theorem, by first embedding it into $\mathbb{R}\mathbb{P}^d$ and then using the dual principle of projective space. This yields:

For any $0<\beta<1$ there is some constand $c(\beta)$ depending on $d$ and $\beta$ such that for any set $S$ of $n$ hyperplanes in $\mathbb{R}\mathbb{P}^d$, either

1. there exists a collection of flats $\{F_1,...,F_k\}$ with $\sum_{i=1}^d codim F_i<d$ such that the sum over the number of hyperplanes which contain at least one of these flats is bigger then $\beta n$

or

2. the number of intersection points from hyperplanes from $S$ exceeds $c(\beta) \cdot n^d$.

Question: Is this the correct dualization or am I messing something up?