Yet another constructive version of Cauchy theorem

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 $f(x)=g(y)$.