I was reading [Hart and Iosevich - Ubiquity of simplices in subsets of vector spaces over finite fields][1] about some quantitative results on simplices in subsets of vector spaces over finite fields. I think that I understood most of the technical details of that paper. However, I did not understand how **Theorem 3.1** implies **Theorem 1.3**.$\newcommand\card[1]{\lvert#1\rvert}\newcommand\norm[1]{\lVert#1\rVert}$

> **Theorem 1.3.** Let $E\subset \mathbb{F}_q^d$, $d>\tbinom{k+1}{2}$ such that $\card E\geq Cq^{\frac{k}{k+1}d+\frac{k}{2}}$ with a
> sufficiently large constant $C>0$. Then $E$ contains an isometric copy
> of every $k$-simplex.
> 
> **Theorem 3.1.** Let $E\subset \mathbb{F}_q^d$, $d>\tbinom{k+1}{2}$, such that $\card E \geq Cq^{\frac{k}{k+1}d+\frac{k}{2}}$ with a
> sufficiently large constant $C>0$. Then for every side length $l_k$,
 $l_k\in (\mathbb{F}_q^{\times})^{\tbinom{k+1}{2}}$ we have
 $\card{\mathcal{T}_{l_k}}>0$. Furthermore, $\card{\mathcal{T}_{l_k}}\sim
 \card E^{k+1}q^{-\tbinom{k+1}{2}}$.
> 
> Using this theorem we recover the main result of the paper using the
> following linear algebraic observation. 
> 
> **Lemma 3.2.** Let $P$ be a simplex with vertices $v_0,v_1,\dotsc, v_k$, $v_j\in \mathbb{F}_q^d$. Let $P'$ be another simplex with vertices
> $v'_0,v'_1,\dotsc,v'_k$. Suppose that $$\norm{v_i-v_j}=\norm{v'_i-v'_j}$$ for all $i,j$. Then there exists an orthogonal,
> affine transformation $O$ on $\mathbb{F}_q^d$ such that $O(P)=P'$.

On page 4 authors write that "*This representation does not, in general, always embody a simplex, as $\mathcal{T}_{l_k}$ is not guaranteed to be in general position. However, as we show below, "legitimate" $k$-simplices are equivalent up to an orthogonal transformation.*"

Question 1: I am really confused how these observations imply Theorem 1.3. 

Question 2: What does "*isometric copy of a $k$-simplex*"  mean?


  [1]: https://arxiv.org/abs/math/0703504