证明：由于：f(x+1)+f(y+1)=f(xy+1)
则有：f(xy+1)-f(y+1)=f(x+1)
任取x1,x2属于(1,正无穷),且x1>x2
则f(x1)-f(x2)
=f[(x1-1)+1]-f[(x2-1)+1]
=f[(x1-1)/(x2-1) +1]
由于:x1>x2>1
则有：x1-1>x2-1>0
故：(x1-1)/(x2-1) >1
则(x1-1)/(x2-1) +1>2
又x>2时,f(x)>0
则：f[(x1-1)/(x2-1) +1] >0
即对任意x1,x2属于(1,正无穷),
当x1>x2时,恒有f(x1)>f(x2)
故f(x)在(1,正无穷)内单调递增