On non-immersibility of $RP^{10}$ to $R^{15}$

B. J. Sanderson noted that for $k < n$ the projective space $RP^k$ is immersible in $R^n$ f and only if the tangent bundle $RP^n$ admits $k$ linearly independent vector fields over $RP^k$ [1, Lemma (9.7)]. Using this remark, P. F. Baum and W. Browder proved that $RP^10$ can not be immersed to $R^{15}$ [1, Corollary (9.9)] by showing that the tangent bundle $TRP^{15}$ does not admit $9$ linearly independent vector fields over $RP^{10}$ [1, Thm. (9.5)]. We present a new proof of this last statement based on U. Koschorke singularity approach [2].
References
[1] P.F. Baum, W.Browder. The cohomology of quotients of classical groups // Topology 3 (1965), 305–336.
[2] U. Koschorke. Vector Fields and Other Vector Bundle Morphisms — A Singularity Approach. Lecture Notes in Math. 847 (Springer, Berlin, 1981).