Given the natural numbers N with the properties of associativity and commutivity of both addition and multiplication and the distributive law with trichotomy(a<b, a>b or a=b) and transitivity(a > b, b > c \Rightarrow a > c) and a < a + c and a < b \Rightarrow ac < bc for all a, b, c in N. With the integers Z defined as the set of ordered pairs (x,y) where x,y \in N and (a,b) < (c,d) \Leftrightarrow a+d < c+b. 0 is defined as equivalence class of (m,m) m \in N and (a+b,b) \in N.How does one show that if z_1, z_2, z_3 \in Z, z_1 < z_2 \Rightarrow z_1 + z_3 < z_2 + z_3 ? My difficulty comes in how does one show that if c+b < a+d \Rightarrow (a+b,c+d) \in N?