Lower bounds on the formula complexity of a linear Boolean function

Given proof of the lower boundaries of the computational complexity of the linear Boolean function $x_1+\ldots+x_n=1 \pmod 2$ by formulas in the basis $\{\vee,\wedge,^-\}$. It is proved that for $n=6$ this complexity is not less than 40. Earlier, this result was obtained Cherukhin with use of computer calculations [1]. Given a simplified proof of the lower bound, published at [2]: for even $n\neq2^k$ the complexity is not less than $n^2+2$, for odd $n\geq5$ the complexity is not less than $n^2+3$.