I have given real numbers x 1 , x 2 , y 1 , y 2 such that x 1 &gt; x 2 and y 1 &lt; y 2 . The the claim is that there exists some λ ∈ ( 0 , 1 ) such that λ ( x 1 − x 2 ) + ( 1 − λ ) ( 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). Is there any constructive way to show the above equation?

Question

I have given real numbers       x    1    ,      x    2    ,      y    1    ,      y    2   such that       x    1    &amp;gt;      x    2   and       y    1    &amp;lt;      y    2  . The the claim is that there exists some   λ  ∈  (  0  ,  1  ) such that   λ  (      x    1    −      x    2    )  +  (  1  −  λ  )  (      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). Is there any constructive way to show the above equation?

Cristal Obrien · Accepted Answer

Just solve the equation for   λ. You get   λ  =                    y        2            −              y        1                            x        1            −              x        2            +              y        2            −              y        1            .

Annabel Sullivan · Accepted Answer

You can explicitly solve the equation;  λ  =                    y        2            −              y        1                            x        1            −              x        2            −              y        1            +              y        2              .

I have given real numbers x 1 </msub> , x 2 </msub> ,

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