
theorem fi2:
for F being preordered Field,
    P being Preordering of F,
    a,b being non zero Element of F
st a <=P, 0.F & b <=P, 0.F holds a <=P, b iff b" <=P, a"
proof
let F be preordered Field, P be Preordering of F,
    a,b be non zero Element of F;
assume AS: a <=P, 0.F & b <=P, 0.F;
Y: a <> 0.F & b <> 0.F;
-a <> -0.F & -b <> -0.F;
then -a is non zero & -b is non zero;
then -((-a)") <=P, -0.F & -((-b)") <=P, -0.F by AS,REALALG1:27;
then X: a" <=P, -0.F & b" <=P, -0.F by YZ;
hereby assume a <=P, b;
   then b * a" <=P, a * a" by X,c55;
   then b * a" <=P, 1.F by Y,VECTSP_1:def 10;
   then 1.F * b" <= P, (b * a") * b" by X,c55;
   then b" <= P, (b" * b) * a" by GROUP_1:def 3;
   then b" <= P, 1.F * a" by Y,VECTSP_1:def 10;
   hence b" <=P, a";
   end;
assume b" <=P, a";
   then a" * b <=P, b" * b by AS,c55;
   then a" * b <=P, 1.F by Y,VECTSP_1:def 10;
   then 1.F * a <=P, (a" * b) * a by AS,c55;
   then a <=P, (a" * a) * b by GROUP_1:def 3;
   then a <=P, 1.F * b by Y,VECTSP_1:def 10;
   hence a <=P, b;
end;
