Multiplication is a binary operation defined as follows:
We can can also look at the geometric construction of multiplication by using our coordinatization of the affine plane. We can see what multiplication of points on the diagonal looks like.
Figure 5: Multiplication of diagonal points
Proof:
and 
are true
by T2.
is unique. If we are
given a and c, then we have 
.
With a combination of T1 and T3, this has a unique solution.
If we are given b and c, we have 
by T1, and
this has a unique solution by T3.
Proof:
This is true by an application of T1.
Suppose 
. Then 
. But 
and 
. Thus 
implies either 
or b=0.
Suppose a=0 or b=0. Then 
or 
. But These both equal 0, thus