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 $\displaystyle \Rightarrow$ a > c) and a < a + c and a < b $\displaystyle \Rightarrow$ ac < bc for all a, b, c in N. With the integers Z defined as the set of ordered pairs (x,y) where $\displaystyle x,y \in N$ and $\displaystyle (a,b) < (c,d) \Leftrightarrow a+d < c+b$. 0 is defined as equivalence class of (m,m) $\displaystyle m \in N$ and $\displaystyle (a+b,b) \in N$.How does one show that if $\displaystyle 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 $\displaystyle c+b < a+d \Rightarrow (a+b,c+d) \in N$?