An iterated sum formula for a spheroid's homotopy class modulo 2-torsion

Let $X$ be a simply connected pointed space with finitely generated homotopy groups. Let $\Pi_n(X)$ denote the set of all continuous maps $a:I^n\to X$ taking $\partial I^n$ to the basepoint. For $a\in\Pi_n(X)$, let $[a]\in\pi_n(X)$ be its homotopy class. For an open set $E\subset I^n$, let $\Pi(E,X)$ be the set of all continuous maps $a:E\to X$ taking $E\cap\partial I^n$ to the basepoint. For a cover $\Gamma$ of $I^n$, let $\Gamma(r)$ be the set of all unions of at most $r$ elements of $\Gamma$. Put $r=(n-1)!$. We prove that for any finite open cover $\Gamma$ of $I^n$ there exist maps $f_E:\Pi(E,X)\to\pi_n(X)\otimes Z[1/2]$, $E\in\Gamma(r)$, such that $$[a]\otimes1=\sum_{E\in\Gamma(r)} f_E(a|_E)$$ for all $a\in\Pi_n(X)$.