
theorem
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" = b" implies a = b
   proof
   assume a" = b";
   then a * b" = 1.F by Y,VECTSP_1:def 10;
   then (a * b") * b = b;
   then a * (b * b") = b by GROUP_1:def 3;
   then a * 1.F = b by Y,VECTSP_1:def 10;
   hence a = b;
   end;
hence thesis by AS,fi2;
end;
