(1)

\begin{align*} C(x+1,y+1) & =\frac{(x+1)!}{(y+1)!(x-y)!}\\ & =\frac{x+1}{y+1}\frac{x!}{y!(x-y)!}\\ & =\left(\frac{x-y}{y+1}+1\right)\frac{x!}{y!(x-y)!}\\ & =\frac{x!}{(y+1)!(x-y-1)!}+\frac{x!}{y!(x-y)!}\\ & =C(x,y+1)+C(x,y) \end{align*}

(2)

\begin{align*} C\left(x-1,y\right)-C\left(x-1,y-1\right) & =\frac{x-y}{x}C\left(x,y\right)-\frac{y}{x}C\left(x,y\right)\\ & =\frac{x-2y}{x}C\left(x,y\right) \end{align*}