  As an application of the upper triangular technology method of (V.P. Snaith:
{\em Stable homotopy -- around the Arf-Kervaire invariant}; Birkh\"{a}user
Progress on Math. Series vol. 273 (April 2009)) it is shown that there do not
exist stable homotopy classes of $ {\mathbb RP}^{\infty} \wedge {\mathbb
RP}^{\infty}$ in dimension $2^{s+1}-2$ with $s \geq 2$ whose composition with
the Hopf map to $ {\mathbb RP}^{\infty}$ followed by the Kahn-Priddy map gives
an element in the stable homotopy of spheres of Arf-Kervaire invariant one.
