I have given real numbers $x_1,x_2,y_1,y_2$ such that $x_1 > x_2$ and $y_1 < y_2$. The the claim is that there exists some $\lambda \in (0,1)$ such that $\lambda (x_1 - x_2) + (1-\lambda)(y_1-y_2) = 0$. In order to proof this, one needs ( at least in my opinion) the intermediate value theorem. But the intermediate value theorem does not hold in constructive mathematics (that is without the law of excluded middle; or constructive mathematics acts in intuitionistic logic). For a proof of this c.f. this paper. Is there any constructive way to show the above equation?