(related to Proposition: Basic Calculations Involving the Difference Operator)
By hypothesis, $D\subseteq\mathbb R$ is a subset of real numbers, $x, x+1\in D,\lambda\in\mathbb R,$ and $f,g:D\to\mathbb R$ are functions. The statements follow immediately from the definition of the difference operator.
$$\begin{array}{rcl} \Delta (\lambda f)(x)&=&\lambda f(x+1)-\lambda f(x)\\ &=&\lambda(f(x+1)-f(x))\\ &=&\lambda \Delta f(x)\\ \end{array}$$
$$\begin{array}{rcl} \Delta (f\pm g)(x)&=&(f\pm g)(x+1)-(f\pm g)(x)\\ &=&f(x+1)\pm g(x+1)-(f(x)\pm g(x))\\ &=&(f(x+1)-f(x))\pm (g(x+1)-g(x))\\ &=&\Delta f(x)\pm \Delta g(x)\\ \end{array}$$
$$\begin{array}{rcl} \Delta (fg)(x)&=&(fg)(x+1)-(fg)(x)\\ &=&f(x+1)g(x+1)-f(x)g(x)\\ &=&f(x+1)g(x+1)-f(x+1)g(x)+f(x+1)g(x)-f(x)g(x)\\ &=&f(x+1)(g(x+1)-g(x))+g(x)(f(x+1)-f(x))\\ &=&f(x+1)\Delta g(x)+g(x)\Delta f(x)\\ \end{array}$$
