\begin{proof}[Proof of the Visualization Theorem:]
Recall that a cohomology class $c \in H^1(K, A)$ is trivial if and only if
the corresponding principal homogeneous space $C$ to $c$ has a $K$-rational point. Intuitively,
to trivialize $c$, it is enough to consider an extension $L / K$, so that $C$ has an
$L$-rational point. But we can always choose an unramified extension $L / K$, such that
the principal homogeneous space $C$ has at least one $L$-rational point, i.e.
res$_{L / K}(c) = 0$, where res$_{L/K} : H^1(K,A) \ra H^1(L,A)$ is the restriction map on
Galois cohomology. Let $A_L = A \times_K L$ and let $J = \textrm{Res}_{L/K}(A_L)$. Since
$A$ is an abelian variety, then so is $J$. The construction in the proof of Prop.
6.2.2 shows that $J$
has dimension at most $[L : K] \cdot \textrm{dim }A$. Moreover, for any group scheme
$S$ over $K$, there is an isomorphism $A_L(S \times_K L) \simeq J(S)$. Thus, we have an inclusion
$$
A(K) \hookrightarrow A_L(L) \simeq J(K)
$$
Next, the functorial injection $A(K) \hookrightarrow J(K)$ corresponds via Yoneda's lemma
to an injective morphism $\iota : A \ra J$ of groups schemes over $K$. Since this
morphism is proper, then $\iota$ is a closed immersion *** CITE ***. Finally,
one needs to check that $\iota : A \ra J$ makes $c \in H^1(K, A)$ visible. We know that
res$_{L/K}(c) = 0$. But by using Shapiro's lemma and computations, one concludes that
$$
H^1(K, J) \simeq H^1(L, A_L),
$$
and via this isomorphism $i^*(c) \mapsto \textrm{res}_{L/K}(c)$, so
$c \in \textrm{Vis}^{(i)}_J(H^1(K, A))$.