oops ok:
A1: Addition is associative
A2: Addition is commutative
A3: every integer has an identity with respect to addition
A4: every integer has an inverse with respect to addition
A5: Multiplication is associative
A6: Multiplication is commutative
A7: every integer has an identity with respect to multiplication
A8: Distributive Law: x(y+z)=xy+xz
A9: Closure Property: a set A is closed with respect to addition and multiplication if x+y is in A and xy is in A
A10: Trichotomy Law: For every integer x, exactly one of the following is true: x is either in the set of negative integers, set of positive integers, or x=0
You can also use the following properties that I have already proved:
P1: a+b=a+c => b=c
P2: a0=0a=0
P3: (-a)b=a(-b)=-(ab)
P4: -(-a)=a
P5: (-a)(-b)=ab
P6: a(b-c)=ab-ac
P7: (-1)a=-a
P8: (-1)(-1)=1