  In the paper we consider two positive contractions
$T,S:L^{1}(A,\tau)\longrightarrow L^{1}(A,\tau)$ such that $T\leq S$, here
$(A,\t)$ is a semi-finite $JBW$-algebra. If there is an $n_{0}\in\mathbb{N}$
such that $\|S^{n_{0}}-T^{n_{0}}\|<1$. Then we prove that $\|S^{n}-T^{n}\|<1$
holds for every $n\geq n_{0}.$
