Geodesic
Yet another constructive version of Cauchy theorem
Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part IV, Tome 20 (1971), pp. 36-39

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Let $(x::y)$, $x$ and $y$ standing for constructive real numbers, denotes the open interval $(min(x,y),max(x,y))$. The following theorem is proved. Let two constructive functions $f$ and $g$ be defined respectively on segments $[x_1,x_2]$ and $[y_1,y_2]$ and let the intervals $(f(x_1)::f(x_2))$ and $(g(x_1)::g(x_2))$ have a point in common. Then an $x$ from $[x_1,x_2]$ and аn $y$ from $[y_1,y_2]$ can be found so that $f(x)=g(y)$. 
@article{ZNSL_1971_20_a3,
     author = {I. D. Zaslavsky and G. S. Tseitin},
     title = {Yet another constructive version of {Cauchy} theorem},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {36--39},
     publisher = {mathdoc},
     volume = {20},
     year = {1971},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1971_20_a3/}
}
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I. D. Zaslavsky; G. S. Tseitin. Yet another constructive version of Cauchy theorem. Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part IV, Tome 20 (1971), pp. 36-39. http://geodesic.mathdoc.fr/item/ZNSL_1971_20_a3/