#### Proof of Theorem 1.3.

We start by proving the uniform continuity. Let
${({u}_{i,n}^{0})}_{n\in {\mathbb{N}}_{0}}$ be bounded sequences in ${H}^{1}$ for
$i=1,2$ and set ${C}_{1}:={\mathrm{max}}_{i=1,2}{lim\; sup}_{n\to \mathrm{\infty}}\parallel {u}_{i,n}^{0}\parallel $. Suppose for a
contradiction that

${|{u}_{1,n}^{0}-{u}_{2,n}^{0}|}_{p}\to 0\mathit{\hspace{1em}}\text{as}n\to \mathrm{\infty}$(3.2)

and that there is ${C}_{2}>0$ such that

${\left|\mathcal{\mathcal{F}}({u}_{1,n}^{0})-\mathcal{\mathcal{F}}({u}_{2,n}^{0})\right|}_{\nu}\ge {C}_{2}\mathit{\hspace{1em}}\text{for all}n.$(3.3)

Successively we will define infinitely many sequences
${({a}_{n}^{k})}_{n}\subseteq {\mathbb{Z}}^{N}$ and ${({u}_{i,n}^{k})}_{n}\subseteq {H}^{1}$,
$i=1,2$, indexed by $k\in {\mathbb{N}}_{0}$ and strictly increasing
functions ${\phi}_{k}:\mathbb{N}\to \mathbb{N}$ with the following
properties:

$\underset{i=1,2}{\mathrm{max}}\underset{n\to \mathrm{\infty}}{lim\; sup}\parallel {u}_{i,n}^{k}\parallel \le {C}_{1},$(3.4)$\underset{n\to \mathrm{\infty}}{lim}{|{u}_{1,n}^{k}-{u}_{2,n}^{k}|}_{p}=0,$(3.5)$\underset{n\to \mathrm{\infty}}{lim\; inf}{\left|\mathcal{\mathcal{F}}({u}_{1,n}^{k})-\mathcal{\mathcal{F}}({u}_{2,n}^{k})\right|}_{\nu}\ge {C}_{2},$(3.6)$\underset{n\to \mathrm{\infty}}{\mathrm{w}-\mathrm{lim}}\left(-{a}_{{\psi}_{\mathrm{\ell}}^{k-1}(n)}^{\mathrm{\ell}}\right)\star {u}_{i,n}^{k}=0\hspace{1em}\text{in}{H}^{1}\text{, if}0\le \mathrm{\ell}k\text{, for}i=1,2\text{,}$(3.7)

and

$\underset{n\to \mathrm{\infty}}{lim}\left|{a}_{{\psi}_{m}^{\mathrm{\ell}}(n)}^{m}-{a}_{n}^{\mathrm{\ell}}\right|=\mathrm{\infty}\mathit{\hspace{1em}}\text{if}0\le m\mathrm{\ell}k.$(3.8)

Here,

${\psi}_{\mathrm{\ell}}^{k}:={\phi}_{\mathrm{\ell}+1}\circ {\phi}_{\mathrm{\ell}+2}\circ \mathrm{\dots}\circ {\phi}_{k}\hspace{1em}\text{if}\mathrm{\ell}=-1,0,1,\mathrm{\dots},k-1$${\psi}_{k}^{k}:={\mathrm{id}}_{\mathbb{N}}.$

We need to say something about the extraction of subsequences.
In order to obtain ${\phi}_{k}$, ${({a}_{n}^{k})}_{n}$, and
${({u}_{i,n}^{k+1})}_{n}$ from $({u}_{i,n}^{k})$, we first pass to a
subsequence ${({u}_{i,{\phi}_{k}(n)}^{k})}_{n}$ of ${({u}_{i,n}^{k})}_{n}$ and
then use its terms in the construction. Once the new sequences
${({a}_{n}^{k})}_{n}$ and ${({u}_{i,n}^{k+1})}_{n}$ are built we may remove a
finite number of terms at their start, modifying ${\phi}_{k}$
accordingly, with the goal of obtaining additional properties.
Beginning with the following iteration there are no more
retrospective changes to the sequences already built. This is
to assure a well defined infinite sequence of sequences, from
which eventually we take the diagonal sequence. In this
setting it seems clearer to make the selection of subsequences
explicit, contrary to what is usually done when using
concentration compactness methods
[16, 14, 15] or when proving a
variational splitting lemma.

For $k=0$ properties (3.4)–(3.8) are
fulfilled by the definition of ${C}_{1}$ and by (3.2) and
(3.3). Assume now that (3.4)–(3.8) hold
for some $k\in {\mathbb{N}}_{0}$. Denote by ${W}_{k}$ the set of $v\in {H}^{1}$
such that there are a sequence $({a}_{n})\subseteq {\mathbb{Z}}^{N}$ and a
subsequence of $({u}_{1,n}^{k})$ with

$\underset{n\to \mathrm{\infty}}{\mathrm{w}-\mathrm{lim}}{a}_{n}\star {u}_{1,n}^{k}=v\mathit{\hspace{1em}}\text{in}{H}^{1}.$

If $\underset{n\to \mathrm{\infty}}{\mathrm{w}-\mathrm{lim}}{a}_{n}\star {u}_{1,n}^{k}=0$ in ${H}^{1}$ were true
for all sequences $({a}_{n})\subseteq {\mathbb{Z}}^{N}$, by
Lemma 3.1 it would follow that ${lim}_{n\to \mathrm{\infty}}{u}_{1,n}^{k}=0$ in ${L}^{p}$. Equation (3.5) and the
continuity of $\mathcal{\mathcal{F}}$ on ${L}^{p}$ would lead to a contradiction with
(3.6). Therefore,

${q}_{k}:=\underset{v\in {W}_{k}}{sup}\parallel v\parallel \in (0,{C}_{1}].$

Pick ${v}^{k}\in {W}_{k}$ such that

$\parallel {v}^{k}\parallel \ge \frac{{q}_{k}}{2}>0.$(3.9)

There are ${({a}_{n}^{k})}_{n}\subseteq {\mathbb{Z}}^{N}$ and a strictly increasing
function ${\phi}_{k}:\mathbb{N}\to \mathbb{N}$ such that

$\underset{n\to \mathrm{\infty}}{\mathrm{w}-\mathrm{lim}}(-{a}_{n}^{k})\star {u}_{1,{\phi}_{k}(n)}^{k}={v}^{k}\mathit{\hspace{1em}}\text{in}{H}^{1}.$

By (3.5) and by
Theorem 2.1
(b) and (c)
there exists a sequence ${({v}_{n}^{k})}_{n}\subseteq {H}^{1}$ such that

$\underset{n\to \mathrm{\infty}}{lim}{v}_{n}^{k}={v}^{k}\hspace{1em}\text{in}{H}^{1}\text{,}$(3.10)$\underset{n\to \mathrm{\infty}}{\mathrm{w}-\mathrm{lim}}(-{a}_{n}^{k})\star {u}_{i,{\phi}_{k}(n)}^{k}={v}^{k}\hspace{1em}\text{in}{H}^{1}\text{, for}i=1,2,$(3.11)

and

$\underset{n\to \mathrm{\infty}}{lim}{\left|\mathcal{\mathcal{F}}\left((-{a}_{n}^{k})\star {u}_{i,{\phi}_{k}(n)}^{k}\right)-\mathcal{\mathcal{F}}\left((-{a}_{n}^{k})\star {u}_{i,{\phi}_{k}(n)}^{k}-{v}_{n}^{k}\right)-\mathcal{\mathcal{F}}({v}^{k})\right|}_{\nu}=0,i=1,2.$

Set

${u}_{i,n}^{k+1}:={u}_{i,{\phi}_{k}(n)}^{k}-{a}_{n}^{k}\star {v}_{n}^{k}.$

By the equivariance of $\mathcal{\mathcal{F}}$ and the invariance of the involved
norms under the ${\mathbb{Z}}^{N}$-action,

$\underset{n\to \mathrm{\infty}}{lim}{\left|\mathcal{\mathcal{F}}({u}_{i,{\phi}_{k}(n)}^{k})-\mathcal{\mathcal{F}}({u}_{i,n}^{k+1})-\mathcal{\mathcal{F}}({a}_{n}^{k}\star {v}^{k})\right|}_{\nu}=0\mathit{\hspace{1em}}\text{for}i=1,2\text{,}$(3.12)

and, since by (3.11) the map ${\parallel \cdot \parallel}^{2}$ BL-splits along
$(-{a}_{n}^{k})\star {u}_{i,{\phi}_{k}(n)}^{k}$ with respect to ${v}^{k}$, we have

$\underset{n\to \mathrm{\infty}}{lim}\left|{\parallel {u}_{i,{\phi}_{k}(n)}^{k}\parallel}^{2}-{\parallel {u}_{i,n}^{k+1}\parallel}^{2}-{\parallel {v}^{k}\parallel}^{2}\right|=0\mathit{\hspace{1em}}\text{for}i=1,2\text{.}$(3.13)

Equations (3.13) and (3.4) (for *k*) imply that

$\underset{i=1,2}{\mathrm{max}}\underset{n\to \mathrm{\infty}}{lim\; sup}\parallel {u}_{i,n}^{k+1}\parallel \le {C}_{1},$

hence (3.4) for $k+1$. The
definition of the sequences ${u}_{i,n}^{k+1}$ and (3.5)
(for *k*) imply that

$\underset{n\to \mathrm{\infty}}{lim}{\left|{u}_{1,n}^{k+1}-{u}_{2,n}^{k+1}\right|}_{p}=\underset{n\to \mathrm{\infty}}{lim}{\left|{u}_{1,{\phi}_{k}(n)}^{k}-{u}_{2,{\phi}_{k}(n)}^{k}\right|}_{p}=0,$(3.14)

hence (3.5) for $k+1$. It follows from (3.12)
and (3.6) (for *k*) that

$\underset{n\to \mathrm{\infty}}{lim\; inf}{\left|\mathcal{\mathcal{F}}({u}_{1,n}^{k+1})-\mathcal{\mathcal{F}}({u}_{2,n}^{k+1})\right|}_{\nu}=\underset{n\to \mathrm{\infty}}{lim\; inf}{\left|\mathcal{\mathcal{F}}({u}_{1,{\phi}_{k}(n)}^{k})-\mathcal{\mathcal{F}}({u}_{2,{\phi}_{k}(n)}^{k})\right|}_{\nu}\ge {C}_{2},$(3.15)

hence (3.6) for $k+1$. Last but not least, from
(3.7) (for *k*), (3.9), and (3.11) it
follows that

$\underset{n\to \mathrm{\infty}}{lim}\left|{a}_{{\psi}_{m}^{k}(n)}^{m}-{a}_{n}^{k}\right|=\mathrm{\infty}\mathit{\hspace{1em}}\text{if}mk.$(3.16)

Since (3.8) is true for *k*, together with
(3.16) we obtain (3.8) for $k+1$. Moreover,
(3.16), (3.7) (for *k*) and (3.10) yield

$\underset{n\to \mathrm{\infty}}{\mathrm{w}-\mathrm{lim}}\left(-{a}_{{\psi}_{\mathrm{\ell}}^{k}(n)}^{\mathrm{\ell}}\right)\star {u}_{i,n}^{k+1}=\underset{n\to \mathrm{\infty}}{\mathrm{w}-\mathrm{lim}}\left(\left(-{a}_{{\psi}_{\mathrm{\ell}}^{k-1}({\phi}_{k}(n))}^{\mathrm{\ell}}\right)\star {u}_{i,{\phi}_{k}(n)}^{k}-\left({a}_{n}^{k}-{a}_{{\psi}_{\mathrm{\ell}}^{k}(n)}^{\mathrm{\ell}}\right)\star {v}_{n}^{k}\right)=0\mathit{\hspace{1em}}\text{in}{H}^{1}\text{if}\mathrm{\ell}k.$

By the definition of ${a}_{n}^{k}$, we have

$\underset{n\to \mathrm{\infty}}{\mathrm{w}-\mathrm{lim}}(-{a}_{n}^{k})\star {u}_{i,n}^{k+1}=\underset{n\to \mathrm{\infty}}{\mathrm{w}-\mathrm{lim}}\left((-{a}_{n}^{k})\star {u}_{i,{\phi}_{k}(n)}^{k}-{v}_{n}^{k}\right)=0\mathit{\hspace{1em}}\text{in}{H}^{1}\text{.}$

This proves (3.7) for $k+1$.

We now skip a finite number of elements of the sequences
constructed in this induction step and adapt ${\phi}_{k}$
accordingly. Choosing $m\in \mathbb{N}$ large enough, by (3.14)
and (3.15) we obtain

${\left|{u}_{1,m+n}^{k+1}-{u}_{2,m+n}^{k+1}\right|}_{p}\le \frac{1}{k+1}$

and

${\left|\mathcal{\mathcal{F}}({u}_{1,m+n}^{k+1})-\mathcal{\mathcal{F}}({u}_{2,m+n}^{k+1})\right|}_{\nu}\ge {C}_{2}-\frac{1}{k+1}$

for all $n\in \mathbb{N}$. Property (3.8) (for $k+1$) implies
that

$\underset{n\to \mathrm{\infty}}{lim}\left|{a}_{{\psi}_{m}^{k}(n)}^{m}-{a}_{{\psi}_{\mathrm{\ell}}^{k}(n)}^{\mathrm{\ell}}\right|=\underset{n\to \mathrm{\infty}}{lim}\left|{a}_{{\psi}_{m}^{\mathrm{\ell}}({\psi}_{\mathrm{\ell}}^{k}(n))}^{m}-{a}_{{\psi}_{\mathrm{\ell}}^{k}(n)}^{\mathrm{\ell}}\right|=\mathrm{\infty}\mathit{\hspace{1em}}\text{if}m\mathrm{\ell}\le k.$

Since ${\parallel \cdot \parallel}^{2}$ BL-splits along weakly convergent
sequences, this yields, together with (3.10), that

$\underset{n\to \mathrm{\infty}}{lim}{\parallel \sum _{j=\mathrm{\ell}}^{k}{a}_{{\psi}_{j}^{k}(n)}^{j}\star {v}_{{\psi}_{j}^{k}(n)}^{j}\parallel}^{2}=\sum _{j=\mathrm{\ell}}^{k}{\parallel {v}^{j}\parallel}^{2}$

for all $\mathrm{\ell}\le k$. For large enough *m* this implies

${\parallel \sum _{j=\mathrm{\ell}}^{k}{a}_{{\psi}_{j}^{k-1}({\phi}_{k}(m+n))}^{j}\star {v}_{{\psi}_{j}^{k-1}({\phi}_{k}(m+n))}^{j}\parallel}^{2}\le 2\sum _{j=\mathrm{\ell}}^{k}{\parallel {v}^{j}\parallel}^{2}\mathit{\hspace{1em}}\text{for all}n\in \mathbb{N}\text{and}\mathrm{\ell}\le k.$

We fix *m* with these properties, we write ${u}_{i,n}^{k+1}$,
${a}_{n}^{k}$, and ${v}_{n}^{k}$ instead of ${u}_{i,m+n}^{k+1}$, ${a}_{m+n}^{k}$,
and ${v}_{m+n}^{k}$, respectively, and we write ${\phi}_{k}(n)$
instead of ${\phi}_{k}(m+n)$, thus all properties proved above remain
valid, and, in addition, the following hold true:

${\left|{u}_{1,n}^{k+1}-{u}_{2,n}^{k+1}\right|}_{p}\le \frac{1}{k+1}$(3.17)

and

${\left|\mathcal{\mathcal{F}}({u}_{1,n}^{k+1})-\mathcal{\mathcal{F}}({u}_{2,n}^{k+1})\right|}_{\nu}\ge {C}_{2}-\frac{1}{k+1}$(3.18)

for all $n\in \mathbb{N}$ and

${\parallel \sum _{j=\mathrm{\ell}}^{k}{a}_{{\psi}_{j}^{k}(n)}^{j}\star {v}_{{\psi}_{j}^{k}(n)}^{j}\parallel}^{2}\le 2\sum _{j=\mathrm{\ell}}^{k}{\parallel {v}^{j}\parallel}^{2}\mathit{\hspace{1em}}\text{for all}n\in \mathbb{N}\text{and}\mathrm{\ell}\le k.$(3.19)

Now we consider the process of constructing sequences as
finished and proceed to prove properties of the whole set.
By induction, (3.13) leads to

${\parallel {u}_{1,n}^{k+1}\parallel}^{2}={\parallel {u}_{1,{\psi}_{-1}^{k}(n)}^{0}\parallel}^{2}-\sum _{j=0}^{k}{\parallel {v}^{j}\parallel}^{2}+o(1)\mathit{\hspace{1em}}\text{as}n\to \mathrm{\infty},$

and hence ${\sum}_{j=0}^{\mathrm{\infty}}{\parallel {v}^{j}\parallel}^{2}\le {C}_{1}$ by
(3.4). In view of (3.9) this yields

${q}_{k}\to 0\mathit{\hspace{1em}}\text{as}k\to \mathrm{\infty}.$(3.20)

We claim that the
diagonal sequence $({u}_{1,n}^{n})$ satisfies

${b}_{n}\star {u}_{1,n}^{n}\rightharpoonup 0\mathit{\hspace{1em}}\text{in}{H}^{1}\text{as}n\to \mathrm{\infty}\text{, for every sequence}({b}_{n})\subseteq \mathbb{Z}\text{.}$(3.21)

Note that by construction, for all $\mathrm{\ell}\le k$ we have

${u}_{1,n}^{k}={u}_{1,{\psi}_{\mathrm{\ell}-1}^{k-1}(n)}^{\mathrm{\ell}}-\sum _{j=\mathrm{\ell}}^{k-1}{a}_{{\psi}_{j}^{k-1}(n)}^{j}\star {v}_{{\psi}_{j}^{k-1}(n)}^{j}.$

Hence we have the representation

${u}_{1,n}^{n}={u}_{1,{\psi}_{k-1}^{n-1}(n)}^{k}-\sum _{j=k}^{n-1}{a}_{{\psi}_{j}^{n-1}(n)}^{j}\star {v}_{{\psi}_{j}^{n-1}(n)}^{j}\mathit{\hspace{1em}}\text{if}n\ge k.$(3.22)

First we show that

$\underset{n\to \mathrm{\infty}}{\mathrm{w}-\mathrm{lim}}(-{a}_{{\psi}_{k}^{n-1}(n)}^{k})\star {u}_{1,n}^{n}=0\mathit{\hspace{1em}}\text{in}{H}^{1}\text{, for all}k\in {\mathbb{N}}_{0}\text{.}$(3.23)

Fix $k\in {\mathbb{N}}_{0}$. For every $w\in {H}^{1}$ and $\epsilon >0$
there is ${\mathrm{\ell}}_{0}\ge k+1$ such that

${\parallel w\parallel}^{2}\sum _{j={\mathrm{\ell}}_{0}}^{\mathrm{\infty}}{\parallel {v}^{j}\parallel}^{2}\le {\epsilon}^{2}/2.$

Then (3.19), (3.22), and the translation
invariance of the norm yield for $n\ge {\mathrm{\ell}}_{0}$ the following:

$\left|\u3008\left(-{a}_{{\psi}_{k}^{n-1}(n)}^{k}\right)\star {u}_{1,n}^{n},w\u3009\right|\le \left|\u3008\left(-{a}_{{\psi}_{k}^{n-1}(n)}^{k}\right)\star {u}_{1,{\psi}_{k}^{n-1}(n)}^{k+1},w\u3009\right|+\left|\u3008{\displaystyle \sum _{j=k+1}^{{\mathrm{\ell}}_{0}-1}}\left({a}_{{\psi}_{j}^{n-1}(n)}^{j}-{a}_{{\psi}_{k}^{n-1}(n)}^{k}\right)\star {v}_{{\psi}_{j}^{n-1}(n)}^{j},w\u3009\right|$$+\parallel w\parallel \parallel {\displaystyle \sum _{j={\mathrm{\ell}}_{0}}^{n-1}}{a}_{{\psi}_{j}^{n-1}(n)}^{j}\star {v}_{{\psi}_{j}^{n-1}(n)}^{j}\parallel $$\le \left|\u3008\left(-{a}_{{\psi}_{k}^{n-1}(n)}^{k}\right)\star {u}_{1,{\psi}_{k}^{n-1}(n)}^{k+1},w\u3009\right|+\left|\u3008{\displaystyle \sum _{j=k+1}^{{\mathrm{\ell}}_{0}-1}}\left({a}_{{\psi}_{j}^{n-1}(n)}^{j}-{a}_{{\psi}_{k}^{n-1}(n)}^{k}\right)\star {v}_{{\psi}_{j}^{n-1}(n)}^{j},w\u3009\right|+\epsilon .$

It is easy to see that the sequence ${({\psi}_{k}^{n-1}(n))}_{n}$ is
strictly increasing. Hence the first term in the last
expression tends to 0 as $n\to \mathrm{\infty}$ by (3.7), and
the second term tends to 0 by (3.10) and
(3.16). Since $\epsilon >0$ and $w\in {H}^{1}$ were
arbitrary, this proves (3.23).

To finish the proof of (3.21), suppose for a
contradiction that $\underset{n\to \mathrm{\infty}}{\mathrm{w}-\mathrm{lim}}{b}_{n}\star {u}_{1,n}^{n}=v\ne 0$ in ${H}^{1}$, for a subsequence. Equation (3.23)
implies that

$\underset{n\to \mathrm{\infty}}{lim}\left|{b}_{n}+{a}_{{\psi}_{k}^{n-1}(n)}^{k}\right|=\mathrm{\infty}$

for every $k\in {\mathbb{N}}_{0}$. Pick $k\in {\mathbb{N}}_{0}$ such that
${q}_{k}<\parallel v\parallel $. This is possible by (3.20). Then,
for every $w\in {H}^{1}$, it follows from (3.19) and
(3.22) that

$\left|\u3008{b}_{n}\star {u}_{1,{\psi}_{k-1}^{n-1}(n)}^{k}-v,w\u3009\right|\le \left|\u3008{b}_{n}\star {u}_{1,n}^{n}-v,w\u3009\right|+\left|\u3008\sum _{j=k}^{n-1}\left({b}_{n}+{a}_{{\psi}_{j}^{n-1}(n)}^{j}\right)\star {v}_{{\psi}_{j}^{n-1}(n)}^{j},w\u3009\right|\to 0$

as $n\to \mathrm{\infty}$, as above. Hence,

$\underset{n\to \mathrm{\infty}}{\mathrm{w}-\mathrm{lim}}\left({b}_{n}\star {u}_{1,{\psi}_{k-1}^{n-1}(n)}^{k}\right)=v$

with $\parallel v\parallel >{q}_{k}$. Since

${\left({u}_{1,{\psi}_{k-1}^{n-1}(n)}^{k}\right)}_{n}$

is a subsequence of
${({u}_{1,n}^{k})}_{n}$, this contradicts the definition of ${q}_{k}$ and
proves (3.21).

We are now in the position to finish the proof of uniform
continuity of $\mathcal{\mathcal{F}}$. Equations (3.17) and (3.18) imply that

$\underset{n\to \mathrm{\infty}}{lim}{|{u}_{1,n}^{n}-{u}_{2,n}^{n}|}_{p}=0$(3.24)

and

$\underset{n\to \mathrm{\infty}}{lim\; inf}{\left|\mathcal{\mathcal{F}}({u}_{1,n}^{n})-\mathcal{\mathcal{F}}({u}_{2,n}^{n})\right|}_{\nu}\ge {C}_{2}.$(3.25)

By Lemma 3.1 and (3.21), we have ${u}_{1,n}^{n}\to 0$ in
${L}^{p}$. Together with (3.24) and (3.25) this
contradicts the continuity of $\mathcal{\mathcal{F}}$ on ${L}^{p}$ and therefore
proves the assertion about uniform continuity.

It only remains to prove BL-splitting for $\mathcal{\mathcal{F}}$ along weakly
convergent sequences in ${H}^{1}$ with respect to their weak
limits. Suppose that ${u}_{n}\rightharpoonup v$ in ${H}^{1}$. By
Theorem 2.1
(b) there is a
sequence $({v}_{n})\subseteq {H}^{1}$ such that ${v}_{n}\to v$ in ${H}^{1}$
and, after passing to a subsequence of $({u}_{n})$, we have

$\mathcal{\mathcal{F}}({u}_{n})-\mathcal{\mathcal{F}}({u}_{n}-{v}_{n})\to \mathcal{\mathcal{F}}(v)\mathit{\hspace{1em}}\text{in}{L}^{\nu}$(3.26)

as $n\to \mathrm{\infty}$. Since $({u}_{n})$ and $({v}_{n})$ are bounded in
${H}^{1}$, and by the uniform continuity of $\mathcal{\mathcal{F}}$ on bounded
subsets of ${H}^{1}$ with respect to the ${L}^{p}$-norm (and hence also
with respect to the ${H}^{1}$-norm), it follows that we may replace
${v}_{n}$ by *v* in (3.26). Using this, a standard
reasoning by contradiction yields the claim.
∎

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