We have the following set of axioms for $a, b \in \mathbb{Z}$:
$I1: a+b$ and $ab$ $\in \mathbb{Z}$
$I2:$ $a+b=b+a$ and $ab=ba$
$I3: (a+b)+c=a+(b+c)$ and $a(bc)=(ab)c$
$I4:$ There are numbers $0$, $1$ whith $0\not=1$ such that $a+0=a$ and $a1=a$
$I5:$ $a(b+c)=ab+ac$
$I6:$ For each $a \in \mathbb{Z}$ there is a unique number $-a\in\mathbb{Z}$ such that $a+(-a)=0$
$I7:$ If $a\not=0$ and $ab=ac$ then $b=c$
These axioms are pretty basic. The point is that, in my discrete mathematics course, we were asked to forget everything we knew about mathematics to start building up all the operations and theorems from the given set of axioms. That is why the definition of $x^2$ is explicitly given on the excercise, for example.
Apart from this axioms, I'm allowed to use these other properties that I have proven already from the axioms, such as:
$*-(-a)=a$, $(-a)(-b)=ab$
$*a(-b)=-a(b)=-(ab)$
$*0x=0$ for any $x$.
$Problem:$ For $a, b \in \mathbb{Z}$ prove that there is one and only one $c \in \mathbb{Z}$ such that $(a+b)c=a^2-b^2$, where $a+b \not=0$ and $x^2=xx.$
I have failed at every attempt to prove this. I am not allowed to use anything that is not defined on the axioms or proven properties, such as division. Only sum, difference and product are defined operations. I would show my attempts, but they fail too quickly to give any glance at where I'm struggling, since I really wasn't able to advance much. Any ideas on how to prove these?