According to the concept that R-{0} is a group of nonempty sets (1), it is proved that any A belongs to R-{0} and any B does not belong to R-{0}, so a*b! = 0, a*b is a real number. A*b belongs to R-{0} (2)(a*b)*c = a*(b*c) satisfies the associative law. (3) There is a real number. E = 1 belongs to R-{0} satisfaction1* a = a *1= a *1. A (- 1) = 1/A belongs to R-{0} so that a *1/a =1has an inverse. In a word, R-{0} is a group. Where 1 is the identity of R-{0} and 1/a is the inverse of a.