Show identity of subgroup is same as identity of group

Working in $G$, we have $1_H1_H=1_H=1_H1_G$. The first equality follows from the fact that $1_H$ is the identity of $H$ and $H$ inherits its operation from $G$. The second follows from the fact that $1_G$ is the identity of $G$. Now premultiply by $1_H^{-1}$ to obtain the result.

Hint:

Start with $1_{H}^2 = 1_{H}$.

The identity of $H$ is an element of $G$ satisfying $x^2 = x$, and the only such element can be $1_G$.