$\forall$$k$:$\mathbb{N}$, $f$:(\{0..$k$$^{-}$\}$\rightarrow$\{0..$k$$^{-}$\}). increasing($f$;$k$) $\Rightarrow$ ($\forall$$i$:\{0..$k$$^{-}$\}. $f$($i$) = $i$ $\in$ $\mathbb{Z}$)