
theorem ord5:
for F being preordered Field
for P being Preordering of F
for a being non zero Element of F st a in P holds a" in P
proof
let F be preordered Field; let P be Preordering of F;
let a be non zero Element of F;
assume A: a in P;
E: a <> 0.F;
set b = a";
C: P * P c= P by defppc;
b * b = b^2;
then b * b in P by ord1;
then B: a*(b*b) in {x*y where x,y is Element of F : x in P & y in P} by A;
a * (b * b) = (a * b) * b by GROUP_1:def 3
           .= 1.F * b by E,VECTSP_1:def 10
           .= b;
hence thesis by B,C;
end;
