topological vector spaces – Theorem 2.5 Rudin functional analysis, use of balanced neighbors.


Related to this question (I sometimes read through the book to refresh my background in functional analysis).

The theorem starts with picking balanced neighbours $U$, $W$ such that
$$
bar{U} + bar{U} subset W
$$

But the conclusion of the theorem is reached through the chains of inclusions

$$
Lambda(V) subset Lambda x – Lambda (E) subset bar{U} – bar{U} subset W
$$

I’m comparing $bar{U} + bar{U} subset W$ against $bar{U} – bar{U} subset W$. In chapter 1 the definition of balanced is:

A set $B subset X$ (topological vector space) is said to be balanced if $alpha B subset B$ for every $alpha in Phi$ with $|alpha| leq 1$.

Therefore I suppose we have the inclusion $- bar{U} subset bar{U}$ (since $|-1|leq 1$) and therefore

$$
bar{U} – bar{U} subset bar{U} + bar{U}
$$

However my question is whether or not is $- bar{U} subset bar{U}$ or $- bar{U} = bar{U}$, I think the latter is the definition of symmetric set.