
theorem fi1:
for F being preordered Field,
    P being Preordering of F,
    a,b being non zero Element of F
st 0.F <=P, a & 0.F <=P, b 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: 0.F <=P, a & 0.F <=P, b;
then X: 0.F <=P, a" & 0.F <=P, b" by REALALG1:27;
Y: a <> 0.F & b <> 0.F;
hereby assume a <=P, b;
   then a * a" <=P, b * a" by X,c5;
   then 1.F <=P, b * a" by Y,VECTSP_1:def 10;
   then 1.F * b" <= P, (b * a") * b" by X,c5;
   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 b" * b <=P, a" * b by AS,c5;
   then 1.F <=P, a" * b by Y,VECTSP_1:def 10;
   then 1.F * a <=P, (a" * b) * a by AS,c5;
   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;
