\input amstex

$$\aligned
\frac{1}{x\,(x-y)\,(x-z)}-\frac{1}{y\,(z-y)\,(y-x)}+\frac{1}{z\,(z-x)\,(z-y)}
&=\frac{y\,z\,(y-z)-x\,z\,(x-z)+x\,y\,(x-y)}{x\,y\,z\,(x-y)\,(y-z)\,(x-z)}=\cr
&=\frac{z\,\bigl[y\,(y-z)-x\,(x-z)\bigr]+x\,y\,(x-y)}{x\,y\,z\,(x-y)\,(y-z)\,(x-z)}=\cr
&=\frac{z\,(y^2-yz-x^2+xz)+x\,y\,(x-y)}{x\,y\,z\,(x-y)\,(y-z)\,(x-z)}=\cr
&=\frac{z\,\bigl[(x-y)\,z-(x+y)(x-y)\bigr]+x\,y\,(x-y)}{x\,y\,z\,(x-y)\,(y-z)\,(x-z)}=\cr
&=\frac{(x-y)\bigl[z^2-z\,(x+y)\bigr]+x\,y\,(x-y)}{x\,y\,z\,(x-y)\,(y-z)\,(x-z)}=\cr
&=\frac{(x-y)(z^2-yz-xz+xy)}{x\,y\,z\,(x-y)\,(y-z)\,(x-z)}=\cr
&=\frac{(x-y)(x-z)(y-z)}{x\,y\,z\,(x-y)\,(y-z)\,(x-z)}=\frac{1}{xyz}
\endaligned$$
\bye