$X$ is the smallest set of polynomials $p(x)$ such that: 1. $p(x) = x$ belongs to $X$. 2. If $r(x)$ belongs to $X$, then $x\cdot r(x)$ and $(x + (1 - x) \cdot r(x) )$ both belong to $X$. Show that if $r(x)$ and $s(x)$ are distinct elements of $X$, then $r(x) \neq s(x)$ for any $0 < x < 1$.