[proofplan] We verify the defining properties of a $2$-$(v,k,\lambda)$ design directly. The point set has size $v$, each translate of $D$ has size $k$, and the essential point is to count, for two distinct points $a,b \in G$, how many indexed translates $B_g$ contain both. That count is put in bijection with the ordered representations of the nonzero difference $b-a$ as $y-x$ with $x,y \in D$, which is exactly $\lambda$ by the difference set hypothesis. [/proofplan]

[step:Verify the number of points and the size of every translated block] The point set is $G$, so it has cardinality $|G|=v$. For each $g \in G$, define the translation map $\tau_g: D \to B_g$ by sending each $d \in D$ to $g+d$. The inverse map $B_g \to D$ sends each $z \in B_g$ to $z-g$, so $\tau_g$ is a bijection. Hence \begin{align*} |B_g| = |D| = k \end{align*} for every $g \in G$. [/step]

[step:Count translated blocks through a fixed pair of distinct points]Let $a,b \in G$ be distinct points. Define the set of indices of blocks containing both points by \begin{align*} I(a,b) := \{g \in G : a \in B_g \text{ and } b \in B_g\}. \end{align*} For $g \in G$, the conditions $a \in B_g$ and $b \in B_g$ are equivalent to the existence of elements $x,y \in D$ such that \begin{align*} a = g+x, \qquad b = g+y. \end{align*} For a fixed $g$, these elements are uniquely determined as $x=a-g$ and $y=b-g$. Therefore define the map $\Phi: I(a,b) \to \{(x,y) \in D \times D : b-a = y-x\}$ by sending each $g \in I(a,b)$ to $(a-g,b-g)$. This map is well-defined. It is bijective. Indeed, if $g \in I(a,b)$, then $a-g,b-g \in D$ and \begin{align*} (b-g)-(a-g)=b-a, \end{align*} so $\Phi(g)$ lies in the displayed set. Conversely, if $(x,y) \in D \times D$ satisfies $b-a=y-x$, define $g:=a-x$. Then \begin{align*} g+y = a-x+y = a+(y-x)=a+(b-a)=b, \end{align*} and also $g+x=a$, so $a,b \in B_g$. Thus $g \in I(a,b)$ and $\Phi(g)=(x,y)$. Since $a \ne b$, the element $b-a$ is nonzero. By the difference set hypothesis, there are exactly $\lambda$ ordered pairs $(x,y) \in D \times D$ such that $b-a=y-x$. Hence \begin{align*} |I(a,b)|=\lambda. \end{align*}[/step]

[guided]Fix two distinct points $a,b \in G$. We need to count blocks with multiplicity, so we count indices $g \in G$, not merely distinct subsets of $G$. Define \begin{align*} I(a,b) := \{g \in G : a \in B_g \text{ and } b \in B_g\}. \end{align*} An index $g$ belongs to $I(a,b)$ exactly when both $a$ and $b$ can be written using the same translate $g+D$. That means there are elements $x,y \in D$ such that \begin{align*} a = g+x, \qquad b = g+y. \end{align*} For this fixed $g$, the possible elements of $D$ are forced: subtracting $g$ gives $x=a-g$ and $y=b-g$. Therefore each block index containing both points determines one ordered pair in $D \times D$. Define $\Phi: I(a,b) \to \{(x,y) \in D \times D : b-a = y-x\}$ by sending each $g \in I(a,b)$ to $(a-g,b-g)$. This map is well-defined because $g \in I(a,b)$ implies $a-g \in D$ and $b-g \in D$, and the difference of the two entries is \begin{align*} (b-g)-(a-g)=b-a. \end{align*} Now take any ordered pair $(x,y) \in D \times D$ satisfying $b-a=y-x$. We recover the translate index by defining $g:=a-x$. Then \begin{align*} g+x=a \end{align*} and \begin{align*} g+y=a-x+y=a+(y-x)=a+(b-a)=b. \end{align*} Thus $a,b \in B_g$, so $g \in I(a,b)$, and $\Phi(g)=(x,y)$. This proves that $\Phi$ is a bijection between block indices containing both $a$ and $b$ and ordered representations of $b-a$ as $y-x$ with $x,y \in D$. Because $a \ne b$, the group element $b-a$ is nonzero. The defining property of a $(v,k,\lambda)$ difference set says that every nonzero element of $G$ has exactly $\lambda$ such ordered representations. Therefore \begin{align*} |I(a,b)|=\lambda. \end{align*}[/guided]

[step:Conclude the incidence structure is a $2$-$(v,k,\lambda)$ design] We have shown that the point set has size $v$, every indexed block $B_g$ has size $k$, and every pair of distinct points $a,b \in G$ is contained in exactly $\lambda$ indexed blocks. Since $\mathcal{A}=(B_g)_{g \in G}$ is treated as a multiset indexed by $G$, coincident translates are counted according to their number of indices. These are precisely the defining properties of a $2$-$(v,k,\lambda)$ design with repeated blocks allowed. [/step]