is [strictly quasimonotone]{}.

First of all observe, that for every $x,y\in D(T),\,x\neq y$ there exists $z\in(x,y)$ and $z^*\in T(z)$ such that $\<z^*,y-x\>\neq0.$ Indeed, if $[x,y]\cap D$ is dense in $[x,y]$ then the statement follows from the local strict quasimonotonicity property of $T$ on $D.$ Otherwise, there exists $z\in(x,y),\,z\not\in D$. But then the statement follows from condition (\[e1\]). Hence, it is enough to show, that for all $(x,x^*),(y,y^*)\in G(T),$ such that $\<x^*,y-x\>>0$ we have $\<y^*,y-x\>\ge0.$

We divide the proof into two cases.

If $x,y\in D$ then consider the real set-valued function $$F_{x,y}:[0,1]\rightrightarrows{\mathbb{R}},\,F_{x,y}(t)=\{\<z^*,y-x\>:z^*\in T(x+t(y-x))\}.$$

Obviously $T$ is lower semicontinuous on $[x,y].$ Since $T$ is locally strict quasimonotone on $D$, obviously $T$ is locally strict quasimonotone on $[x,y]\cap D.$ On the other hand $D$ is self segment-dense in $D(T)$, hence $[x,y]\cap D$ is dense in $[x,y].$ Let $$J=\{t\in[0,1]:x+t(y-x)\in [x,y]\cap D\}.$$ Obviously $J$ is dense in $[0,1]$ and $F_{x,y}$ is locally strict quasimonotone on $J$. On the other hand, from \[e1\] we obtain, that $0\not\in F_{x,y}(s)$ for every $s\in[0,1]\setminus J.$ Hence, according to Theorem \[t31\], $F_{x,y} $ is strictly quasimonotone on $[0,1].$ In particular, the latter shows, that for $t^*\in
F_{x,y}(0)$ one has $t^*>0$ then $s^*\ge0$ for all $s^*\in F_{x,y}(1)$, or equivalently if $\<x^*,y-x\>>0$ 