There is nothing to prove: (i) A negative number is defined as the product of any non-zero positive real number. (ii) The product of any number by zero is zero. This is well-defined. -0 means -1 * 0.  (iii) Since -1 is a negative number and the by (ii) the product of any number by zero is zero, it follows that -1 * 0 = 0. But -1 * 0 = -0, therefore -0 = 0.