
theorem FF1:
for F being formally_real Field,
    E being FieldExtension of F
for a being Element of F,
    b being Element of E
st b^2 = a & a in QS F holds FAdj(F,{b}) is formally_real
proof
let F be formally_real Field, E be FieldExtension of F;
let a be Element of F, b be Element of E;
assume A: b^2 = a & a in QS F; then
I: b^2 in F;
set P = the Ordering of F;
assume B: not FAdj(F,{b}) is formally_real;
C: now assume P extends_to FAdj(F,{b});
   then consider O being Subset of FAdj(F,{b}) such that
   D: P c= O & O is positive_cone;
   FAdj(F,{b}) is ordered by D;
   hence contradiction by B;
   end;
QS F c= P by REALALG1:24;
hence thesis by A,C,I,oext1;
end;
