Two Non-Negative Integers have Zeckendorf Representations of which one is Shifted Representation of the Other

Theorem
Let $m, n \in \Z_{\ge 0}$ be non-negative integers.

Then there exists a unique set of integers:
 * $\left\{ {k_1, k_2, \ldots, k_r}\right\}$

where:
 * $k_1 \gg k_2 \gg \cdots \gg k_r$

where $a \gg b$ denotes that $a - b > 1$

such that:
 * $m = F_{k_1} + F_{k_2} + \cdots + F_{k_r}$

and:
 * $n = F_{k_1 + 1} + F_{k_2 + 1} + \cdots + F_{k_r + 1}$

Note that:
 * each of the $k$'s may be negative

and:
 * $r$ may equal $0$.