$n+1$, satisfying that the set $\{u_1,\ldots,u_{n+1}\}$ is contained in $\mathcal{A}_2(u)$.

\[l pre-minimax\] Let $\mathcal{U}$ be a JB$^*$-triple. Let $u_1,\ldots,u_n$ be a collection of $n$ minimal tripotents in $\mathcal{U}$ ($n\in \mathbb{N}$). Then for every tripotent $e$ in $\mathcal{U}$ there exists $u$ in $\mathcal{U}_2(e)$ a tripotent with $rank(u)\leq n$ such that $$x\in \{ u_1, \ldots, u_n\}^{\perp} \;\;\;\hbox{  for all $x$ in $\mathcal{U}_2(e)\cap \mathcal{U}_0(u)$}.$$

As a first step we shall prove the case $n=1$. Assume we have a minimal tripotent $u_1$ and an arbitrary tripotent $e$ in $\mathcal{U}$. We will make use again of the extreme ray property. If $e\perp u_1$ we consider $u=0$ (or any minimal tripotent in $\mathcal{U}_2(e)$ if can be found). If $e$ and $u_1$ are not orthogonal, we define $u=\frac{P_2(e)u_1}{\|P_2(e)u_1\|}$ whenever $\|P_2(e)u_1\|\neq 0$ and $u=\frac{\{u_1,u_1,e\}} {\|\{u_1,u_1,e\}\|} $ in case $\|P_2(e)u_1\|= 0$, which is a minimal tripotent in $\mathcal{U}_2(e)$ by Lemma \[l extreme ray prop en Peirce 1\] and Proposition \[p generalised extreme ray property\]. Using Peirce arithmetics our thesis follows straightforwardly.

We have shown then that for every minimal tripotent we can find a tripotent $v$ in $\mathcal{U}_2(e)$ of rank smaller than or equal to 1 such that every element in $\mathcal{U}_2(e)$ orthogonal to $v$ is also ortogonal to 