Representation of Ternary Expansions

Theorem
Let $x \in \R$ be a real number.

Let $x$ be represented in base $3$ notation.

While it may be possible for $x$ to have two different such representations, for example:
 * $\dfrac 1 3 = 0.100000 \ldots_3 = 0.022222 \ldots_3$

it is not possible for $x$ be written in more than one way without using the digit $1$.