Subgroup of Index 2 contains all Squares of Group Elements

Theorem
Let $G$ be a group.

Let $H$ be a subgroup of $G$ whose index is $2$.

Then:
 * $\forall x \in G: x^2 \in H$

Proof
By Subgroup of Index 2 is Normal, $H$ is normal in $G$.

Hence the quotient group $G / H$ exists.

Then we have: