In an ordered field, S, with positive elements, P, for all x,y elements of S define x<y if y-x is an element of P,
prove that only one of the following can be true:
x<y,x=y,y<x
Also prove that if x,y,z are arbitrary and x<y and y<z then x<z.
In an ordered field, S, with positive elements, P, for all x,y elements of S define x<y if y-x is an element of P,
prove that only one of the following can be true:
x<y,x=y,y<x
Also prove that if x,y,z are arbitrary and x<y and y<z then x<z.