Zeckendorf Representation of Integer shifted Right

Theorem
Let $f \left({x}\right)$ be the real function defined as:
 * $\forall x \in \R: f \left({x}\right) = \left\lfloor{x + \phi^{-1} }\right\rfloor$

where:
 * $\left\lfloor{\, \cdot \,}\right\rfloor$ denotes the floor function
 * $\phi$ denotes the golden mean.

Let $n \in \Z_{\ge 0}$ be a positive integer.

Let $n$ be expressed in Zeckendorf representation:
 * $n = F_{k_1} + F_{k_2} + \cdots + F_{k_r}$

with the appropriate restrictions on $k_1, k_2, \ldots, k_r$.

Then:
 * $F_{k_1 - 1} + F_{k_2 - 1} + \cdots + F_{k_r - 1} = f \left({\phi^{-1} n}\right)$

Proof
Follows directly from Zeckendorf Representation of Integer shifted Left, substituting $F_{k_j - 1}$ for $F_{k_j}$ throughout.