[antiform]

From the real number axioms:

Let x \in R. Then we have that

  x*0 + x*0 = 0*x + 0*x = (0+0)*x = 0*x = x*0

by commutativity of multiplication, distributivity, the existence of a neutral element 0 for addition, and the commutativity of multiplication once again. Thus, we have that

  x*0 + x*0 = x*0.

For clarity, define

  A = x*0.

Then we have that

  A + A = A
  A + A + (-A)  = A + (-A), by the existence of additive inverses.
  A + (A + (-A)) = (A + (-A)), by associativity of addition
  A + 0 = 0, by the definition of additive inverses
  A = 0, by the definiton of the neutral element 0 of addition

which gives us our desired result.