
theorem
for F being ordered Field,
    P being Ordering of F
for E being FieldExtension of F
for a being Element of F,
    b,c being Element of E st b^2 = a & c^2 = -a
holds P extends_to FAdj(F,{b}) or P extends_to FAdj(F,{c})
proof
let F be ordered Field, P be Ordering of F;
let E be FieldExtension of F, a be Element of F, b,c be Element of E;
assume AS: b^2 = a & c^2 = -a;
H: P \/ (-P) = the carrier of F  by REALALG1:def 15;
I: b^2 in F & c^2 in F by AS;
assume not P extends_to FAdj(F,{b}); then
  not b^2 in P by I,oext1; then
  a in -P by AS,H,XBOOLE_0:def 3; then
  c^2 in --P by AS;
  hence P extends_to FAdj(F,{c}) by I,oext1;
end;
