Chebyshev subspaces of vector-valued functions

It is shown that if on a compact space $Q$ any polynomial $P_N(z)=\sum_1^Na_i\begin{pmatrix}f_{i1}\\\vdots\\f_{is}\end{pmatrix}$, $\sum_1^N|a_i|^2>0$, in a system of continuous vector functions with real coefficients such that $N=n\cdot s$ and $s=2p+1$ has at most $n-1$ zeros, then $Q$ is homeomorphic to a circle or a part of one.