
theorem FF2:
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 & FAdj(F,{b}) is non formally_real holds -a in QS F
proof
let F be formally_real Field, E be FieldExtension of F;
let a be Element of F, b be Element of E;
set P = the Ordering of F;
assume A: b^2 = a & FAdj(F,{b}) is non formally_real;
per cases;
suppose a is square;
  then consider c being Element of F such that B: c^2 = a;
  b = c or b = -c by A,B,FIELD_9:8;
  then b in the carrier of F;
  then {b} c= the carrier of F by TARSKI:def 1;
  then FAdj(F,{b}) == F by FIELD_7:3;
  then FAdj(F,{b}) is ordered by lemPP;
  hence thesis by A;
  end;
suppose B0: a is non square;
  B1: now assume b in F; then
      reconsider c = b as Element of F;
      F is Subring of E by FIELD_4:def 1; then
      c^2 = a by A,FIELD_6:16;
      hence contradiction by B0;
      end;
  -1.FAdj(F,{b}) in QS FAdj(F,{b}) by A,REALALG2:def 3; then
  consider c being Element of FAdj(F,{b}) such that
  B2: c = -1.FAdj(F,{b}) & c is sum_of_squares;
  consider f being FinSequence of FAdj(F,{b}) such that
  B3: Sum f = c & for i being Nat st i in dom f
      ex d being Element of FAdj(F,{b}) st f.i = d^2 by B2;
  H1: F is Subring of E & F is Subring of FAdj(F,{b}) by FIELD_4:def 1; then
  H2: 1.F = 1.FAdj(F,{b}) by C0SP1:def 3;
  per cases;
  suppose f is empty;
    hence thesis by B2,B3;
    end;
  suppose f is non empty; then
    reconsider f as non empty FinSequence of FAdj(F,{b});
    for i being Element of dom f holds f.i is square by B3; then
    reconsider f as quadratic non empty FinSequence of FAdj(F,{b})
      by REALALG2:def 5;
    Sum f = -1.F by B2,B3,H1,H2,FIELD_6:17; then
    Sum f in F; then
    consider g1,g2 being quadratic non empty FinSequence of F such that
    B4: Sum f = Sum g1 + a * Sum g2 by A,B1,maina;
    B5: now assume Sum g2 is zero;
        then Sum g1 = -1.F by B4,B3,B2,H1,H2,FIELD_6:17;
        then -1.F in QS F & QS F c= P by REALALG1:24;
        hence contradiction by REALALG1:26;
        end;
        Sum f = -1.F by B2,B3,H1,H2,FIELD_6:17; then
        -1.F - a * Sum g2
             = Sum g1 + (a * Sum g2 - a * Sum g2) by B4,RLVECT_1:28
            .= Sum g1 + 0.F by RLVECT_1:15; then
        1.F + Sum g1
             = (1.F + -1.F) - a * Sum g2 by RLVECT_1:def 3
            .= 0.F - a * Sum g2 by RLVECT_1:5
            .= (-a) * Sum g2 by VECTSP_1:9; then
     (1.F + Sum g1) * (Sum g2)"
             = (-a) * (Sum g2 * (Sum g2)") by GROUP_1:def 3
            .= (-a) * 1.F by B5,VECTSP_1:def 10;
    hence thesis by B5;
    end;
  end;
end;
