(1)

\begin{align*} H\left(\pm_{1}1\right)\pm_{2}H\left(\pm_{1}1\right) & =\frac{\pm_{1}1+1}{2}\pm_{2}\frac{\pm_{1}1+1}{2}\\ & =\frac{\pm_{2}1+1}{2}\pm_{1}\frac{\pm_{2}1+1}{2}\\ & =H\left(\pm_{2}1\right)\pm_{1}H\left(\pm_{2}1\right) \end{align*} \begin{align*} H\left(\pm_{1}1\right)\pm_{2}H\left(\pm_{1}1\right) & =\frac{\pm_{1}1+1}{2}\pm_{2}\frac{\pm_{1}1+1}{2}\\ & =\frac{1}{2}\left(1\pm_{1}1\right)\left(1\pm_{2}1\right)\\ & =2\frac{1\pm_{1}1}{2}\cdot\frac{1\pm_{2}1}{2}\\ & =2H\left(\pm_{1}1\right)H\left(\pm_{2}1\right) \end{align*}

(2)

\begin{align*} H\left(\pm_{1}1\right)\pm_{2}H\left(\mp_{1}1\right) & =H\left(\pm_{1}1\right)\pm_{2}\left(1-H\left(\pm_{1}1\right)\right)\\ & =H\left(\pm_{1}1\right)\mp_{2}H\left(\pm_{1}1\right)\pm_{2}1\\ & =H\left(\mp_{2}1\right)\pm_{1}H\left(\mp_{2}1\right)\pm_{2}1\\ & =\pm_{2}\left(1-H\left(\mp_{2}1\right)\right)\pm_{1}H\left(\mp_{2}1\right)\\ & =\pm_{2}H\left(\pm_{2}1\right)\pm_{1}H\left(\mp_{2}1\right)\\ & =H\left(\pm_{2}1\right)\pm_{1}H\left(\mp_{2}1\right) \end{align*} \begin{align*} H\left(\pm_{1}1\right)\pm_{2}H\left(\mp_{1}1\right) & =H\left(\pm_{1}1\right)\pm_{2}\left(1-H\left(\pm_{1}1\right)\right)\\ & =H\left(\pm_{1}1\right)\mp_{2}H\left(\pm_{1}1\right)\pm_{2}1\\ & =2H\left(\pm_{1}1\right)H\left(\mp_{2}1\right)\pm_{2}1 \end{align*}

(2)-2

\begin{align*} H\left(\pm_{1}1\right)\pm_{2}H\left(\mp_{1}1\right) & =\frac{\pm_{1}1+1}{2}\pm_{2}\frac{\mp_{1}1+1}{2}\\ & =\frac{\pm_{2}1+1}{2}\pm_{1}\frac{\mp_{2}1+1}{2}\\ & =H\left(\pm_{2}1\right)\pm_{1}H\left(\mp_{2}1\right) \end{align*} \begin{align*} H\left(\pm_{1}1\right)\pm_{2}H\left(\mp_{1}1\right) & =\frac{\pm_{1}1+1}{2}\pm_{2}\frac{\mp_{1}1+1}{2}\\ & =\frac{1}{2}\left(1\pm_{1}1\right)\left(1\mp_{2}1\right)\pm_{2}1\\ & =2H\left(\pm_{1}1\right)H\left(\mp_{2}1\right)\pm_{2}1 \end{align*}

(3)

\begin{align*} H\left(\pm_{1}1\right)\mp_{2}H\left(\mp_{1}1\right) & =\left[H\left(\pm_{1}1\right)\pm_{2}H\left(\mp_{1}1\right)\right]_{\pm_{2}\rightarrow\mp_{2}}\\ & =\left[H\left(\pm_{2}1\right)\pm_{1}H\left(\mp_{2}1\right)\right]_{\pm_{2}\rightarrow\mp_{2}}\\ & =H\left(\mp_{2}1\right)\pm_{1}H\left(\pm_{2}1\right) \end{align*} \begin{align*} H\left(\pm_{1}1\right)\mp_{2}H\left(\mp_{1}1\right) & =\pm_{1}\left\{ H\left(\pm_{2}1\right)\pm_{1}H\left(\mp_{2}1\right)\right\} \\ & =\pm_{1}\left\{ 2H\left(\pm_{1}1\right)H\left(\mp_{2}1\right)\pm_{2}1\right\} \\ & =2H\left(\pm_{1}1\right)H\left(\mp_{2}1\right)\pm_{1}\pm_{2}1 \end{align*}

(3)-2

\begin{align*} H\left(\pm_{1}1\right)\mp_{2}H\left(\mp_{1}1\right) & =\pm_{1}H\left(\pm_{1}1\right)\mp_{2}\mp_{1}H\left(\mp_{1}1\right)\\ & =\pm_{1}\left\{ H\left(\pm_{1}1\right)\pm_{2}H\left(\mp_{1}1\right)\right\} \\ & =\pm_{1}\left\{ H\left(\pm_{2}1\right)\pm_{1}H\left(\mp_{2}1\right)\right\} \\ & =H\left(\mp_{2}1\right)\pm_{1}H\left(\pm_{2}1\right) \end{align*}

(4)

\begin{align*} H\left(\mp_{1}1\right)\pm_{2}H\left(\pm_{1}1\right) & =\left[H\left(\pm_{1}1\right)\pm_{2}H\left(\mp_{1}1\right)\right]_{\pm_{1}\rightarrow\mp_{1}}\\ & =\left[H\left(\pm_{2}1\right)\pm_{1}H\left(\mp_{2}1\right)\right]_{\pm_{1}\rightarrow\mp_{1}}\\ & =H\left(\pm_{2}1\right)\mp_{1}H\left(\mp_{2}1\right) \end{align*} \begin{align*} H\left(\mp_{1}1\right)\pm_{2}H\left(\pm_{1}1\right) & =\pm_{2}\left(H\left(\pm_{2}1\right)\pm_{1}H\left(\mp_{2}1\right)\right)\\ & =\pm_{2}\left(2H\left(\pm_{1}1\right)H\left(\mp_{2}1\right)\pm_{2}1\right)\\ & =-2H\left(\pm_{1}1\right)H\left(\mp_{2}1\right)+1 \end{align*}

(4)-2

\begin{align*} H\left(\mp_{1}1\right)\pm_{2}H\left(\pm_{1}1\right) & =\pm_{2}\left(H\left(\pm_{1}1\right)\pm_{2}H\left(\mp_{1}1\right)\right)\\ & =\pm_{2}\left(H\left(\pm_{2}1\right)\pm_{1}H\left(\mp_{2}1\right)\right)\\ & =H\left(\pm_{2}1\right)\mp_{1}H\left(\mp_{2}1\right) \end{align*}