What is the real number line? It's a set  R  with binary operations "+" and "*"
and a binary relation "<" satisfying

x+y=y+x
(x+y)+z=x+(y+z)
x*y=y*x
(x*y)*z=x*(y*z)
x*(y+z)=x*y+x*z
there is an element "0" with x+0=x for all x
there is an element "1" with x*1=x for all x
there is an element "-1" with 1+(-1)=0
for each x <> 0 there is an element "1/x" with  x*(1/x) = 1

if a>b then a+c > b+c
if a>b and c>0 then a*c > b*c
for all a either a>0 or a<0 or a=0 (only one).

Every nonempty bounded subset S of  R  has a least upper bound.


Show:

For all "a" there is a "b" with  a+b=0.

If a>0 then (-1)*a<0.