(x+y)(x+z)=x^2+xy+xz+yz=x^2+xy+z(x+y)=x^2+xy+z(1/xyz-z)=x^2+xy+1/xy-z^2=xy+1/xy+(x+z)(x-z)=xy+1/xy+(1/xyz-y)(x-z)=xy+1/xy+yz+1/yz-xy-1/xy=yz+1/yz>=2(x,y,z >0)