
theorem
for F being ordered polynomial_disjoint Field,
    P being Ordering of F
for a,b being non square Element of F st b = -a
holds P extends_to FAdj(F,{sqrt a}) or P extends_to FAdj(F,{sqrt b})
proof
let F be ordered polynomial_disjoint Field, P be Ordering of F;
let a,b be non square Element of F;
H: P \/ -P = the carrier of F by REALALG1:def 15;
assume A: b = -a;
per cases;
suppose a in P;
  hence thesis by oext2;
  end;
suppose not a in P;
  then a in -P by H,XBOOLE_0:def 3;
  then -a in --P;
  hence thesis by A,oext2;
  end;
end;
