With the notations of Broyden's theorem, we denote by $M \in M _ { 3 n } ( \mathbb { R } )$ the following block matrix $$M = \left( \begin{array} { c c c } 0 & 0 & I _ { n } + O \\ 0 & 0 & I _ { n } - O \\ - \left( I _ { n } + { } ^ { t } O \right) & - \left( I _ { n } - { } ^ { t } O \right) & 0 \end{array} \right)$$ Using Tucker's theorem, show that there exist positive vectors $x , z _ { 1 } , z _ { 2 } \in \mathbb { R } ^ { n }$ such that $$\left\{ \begin{array} { l } \left( I _ { n } + O \right) x \geq 0 \\ \left( I _ { n } - O \right) x \geq 0 \\ - \left( I _ { n } + { } ^ { t } O \right) z _ { 1 } - \left( I _ { n } - { } ^ { t } O \right) z _ { 2 } \geq 0 \\ z _ { 1 } + \left( I _ { n } + O \right) x > 0 \\ z _ { 2 } + \left( I _ { n } - O \right) x > 0 \\ x - \left( I _ { n } + { } ^ { t } O \right) z _ { 1 } - \left( I _ { n } - { } ^ { t } O \right) z _ { 2 } > 0 \end{array} \right.$$