Properties of Addition
A1. Associativity: If a,b,cZ, then (a+b)+c = a+(b+c).
A2. Commutativity: If a,bZ, then a+b = b+a.
A3. Identity: If aZ, then a+0 = 0+a = a.
A4. Inverses: If aZ, then there exists a unique integer b such that a+b = b+a = 0.
Properties of Multiplication
M1. Associativity: If a,b,cZ, then (a·b)·c = a·(b·c).
M2. Commutativity: If a,bZ, then a·b = b·a.
M3. Identity: If aZ, then a·1 = 1·a = a.
Distributive Law
Addition and multiplication in Z are related by the distributive law which asserts that if a,b,cZ, then a·(b+c) = a·b+a·c.

P0. Let aZ. Then exactly one of the following is true: aP, a = 0, -aP.
P1. If x,yP, then x=yP.
P2. If x,yP, then x·yP.
V. Principle of Mathematical Induction
Let S be a nonempty subset of the set of positive integers P. Assume that (a) 1S and (b) whenever nS we have n+1S. Then S contains all positive integers.
Order
O1. Exactly on of the following holds: x<y, x = y, x>y.
O2. If x<y, y<z, then x<z.
O3. If x<y and z<w, then x+z < y+w.
O4a. If x < y and z > 0 then xz < yz.
O4b. If x < y and z < 0 then xz > yz.

Integer Proposition 1: If b is a positive integer. Then b is greater than or equal to 1.
Integer Corollary 2: Let x and y be integers. If x > y, then x > y+1.
Integer Corollary 3: Let and b positive integers. Then there exists a positive integer n such that a < nb.
Integer Theorem 4: Let a and b be nonzero integers. Then a · b 0.
Integer Proposition 5: Let S be a finite, nonempty subset of Z. Then S has a largest and smallest element.
Integer Theorem 6: Let S be a nonempty (finite or infinite) subset of N. Then S contains a smallest element.
Integer Theorem 7(Division Algorithm): Let a be a nonnegative integer and let b be a positive integer. Then there exist natural numbers q and t such that 0 < r < b and
a = bq + r.