The order of a homotopy invariant in the stable case

Let $X$, $Y$ be cell complexes, let $U$ be an Abelian group, and let $f\colon[X,Y]\to U$ be a homotopy invariant. By definition, the invariant $f$ has order at most $r$ if the characteristic function of the $r$th Cartesian power of the graph of a continuous map $a\colon X\to Y$ determines the value $f([a])$ $\mathbb{Z}$-linearly. It is proved that, in the stable case (that is, when $\operatorname{dim} X<2n-1$, and $Y$ is $(n-1)$-connected for some natural number $n$), for a finite cell complex $X$ the order of the invariant $f$ is equal to its degree with respect to the Curtis filtration of the group $[X,Y]$.
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