rank(a)\leq n-1 \}.$$

Fix an atomic decomposition, $\sum_{i\geq 1} \lambda_i (x) v_i$, of $x$.

Let us consider a finite rank element in $a$ in $\mathcal{U}$ with $rank(a)=n-1$. We denote by $f_a$ the range tripotent of $a$ which is a finite rank tripotent of rank $n-1$ ($f_a=0$ when $n=1$). For every minimal tripotent $v\in \mathcal{U}_0(f_a)$ we have that $a\perp v$ and, by the minimax principle (\[eq minmax principle\]), $$\lambda_{n}(x) \leq \sup\{ \|x\|_{v}: v\in \mathcal{U}_0(f_a) \hbox{ minimal tripotent}\} =$$ $$\sup\{ \|x-a\|_{v}: v\in \mathcal{U}_0(f_a) \hbox{ minimal tripotent}\}\leq \|x-a\|.$$

On the other hand, taking $a=\sum_{i=1}^{n-1}\lambda_i(x)v_i$ ($a=0$ when $n=1$), we have that $\|x-a\|=\lambda_n(x)$ which finishes the proof.

Ky Fan maximum principle {#sec: Ky Fan}
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The classical Ky Fan maximum principle [@Fan49 Theorem 1] states that given an hermitian operator in $B(H)$, where $H$ is a finite dimensional Hilbert space, for every natural $n$ we have that $$\sum_{i=1}^{n} \lambda_i(A)=\max \sum_{i=1}^{n} <Ax_i,x_i>,$$ when $n$ orthogonal vectors $x_i$ ($i=1,\ldots,n$) vary in $H$.

In order to extend this result to weakly compact JB$^*$-triples, we may think, probably dazzled by the minimax principle given in Theorem \[t minmax\], that we only have to change the values $<Ax_i,x_i>$ with the seminorms $\|A\|_{u_i}$ associated to minimal tripotents. The following example highlights 