One of the defining properties of an "ordered field" is that if a< b then, for any number c, a+ c< b+ c.
Now, if you have x< y and w< z, then you can start with x+ w< y+ w from the property I just stated. If then from w< z, w+ y< z+ y. Finally, you use the "transitivity" of inequality, "if a< b and b< c then a< c" to argue that since x+ w< y+ w and w+ y< z+ y, you get x+ w< z+ y= y+ z.