
theorem T2: :: Serre
for F being preordered Field,
    P being Preordering of F,
    a being Element of F
st not -a in P holds P + a * P is Preordering of F
proof
let F be preordered Field, P be Preordering of F, a be Element of F;
assume ASS: not -a in P;
X: 0.F in P & 1.F in P & P + P c= P & P * P c= P &
   P /\ (-P) = {0.F} & SQ F c= P by REALALG1:25,REALALG1:def 14;
set S = P + a * P;
A: S + S c= S
proof let o be object;
  assume o in S + S;
  then consider c,b being Element of F such that
  H1: o = c + b & c in S & b in S;
  consider c1,c2 being Element of F such that
  H2: c = c1 + c2 & c1 in P & c2 in a*P by H1;
  consider b1,b2 being Element of F such that
  H3: b = b1 + b2 & b1 in P & b2 in a*P by H1;
  consider c3 being Element of F such that H4: c2 = a * c3 & c3 in P by H2;
  consider b3 being Element of F such that H5: b2 = a * b3 & b3 in P by H3;
  H6: c3 + b3 in P + P by H4,H5;
  c2 + b2 = a * (c3 + b3) by H4,H5,VECTSP_1:def 3;
  then H7: c2 + b2 in a * P by H6,X;
  H8: c1 + b1 in P + P by H3,H2;
  (c1+b1) + (c2+b2) = ((c1 + b1) + c2) + b2 by RLVECT_1:def 3
                   .= ((c1 + c2) + b1) + b2 by RLVECT_1:def 3
                   .= o by H1,H2,H3,RLVECT_1:def 3;
  hence o in S by H8,H7,X;
  end;
B: S * S c= S
proof let o be object;
  assume o in S * S;
  then consider c,b being Element of F such that
  H1: o = c * b & c in S & b in S;
  consider c1,c2 being Element of F such that
  H2: c = c1 + c2 & c1 in P & c2 in a*P by H1;
  consider b1,b2 being Element of F such that
  H3: b = b1 + b2 & b1 in P & b2 in a*P by H1;
  consider c3 being Element of F such that H4: c2 = a * c3 & c3 in P by H2;
  consider b3 being Element of F such that H5: b2 = a * b3 & b3 in P by H3;
  H6: o
    = c1 * (b1 + a*b3) + (a*c3) * (b1 + a*b3) by H1,H2,H3,H4,H5,VECTSP_1:def 3
   .= (c1 * b1 + c1 * (a * b3)) + (a * c3) * (b1 + a * b3) by VECTSP_1:def 2
   .= (c1 * b1 + c1 * (a * b3)) +
      ((a * c3) * b1 + (a * c3) * (a * b3)) by VECTSP_1:def 2
   .= c1 * b1 + (c1 * (a * b3) +
      ((a * c3) * b1 + (a * c3) * (a * b3))) by RLVECT_1:def 3
   .= c1 * b1 + ((c1 * (a * b3) + (a * c3) * b1) +
      (a * c3) * (a * b3)) by RLVECT_1:def 3
   .= c1 * b1 + ((a * (c1 * b3) + (a * c3) * b1) +
      (a * c3) * (a * b3)) by GROUP_1:def 3
   .= c1 * b1 + ((a * (c1 * b3) + a * (c3 * b1)) +
      (a * c3) * (a * b3)) by GROUP_1:def 3
   .= c1 * b1 + (a * (c1 * b3 + c3 * b1) +
      (a * c3) * (a * b3)) by VECTSP_1:def 2
   .= c1 * b1 + (a * (c1 * b3 + c3 * b1) +
      ((a * c3) * a) * b3) by GROUP_1:def 3
   .= c1 * b1 + (a * (c1 * b3 + c3 * b1) +
      ((a * a) * c3) * b3) by GROUP_1:def 3
   .= c1 * b1 + (a * (c1 * b3 + c3 * b1) +
      (a * a) * (c3 * b3)) by GROUP_1:def 3
   .= (c1 * b1 + (a * a) * (c3 * b3)) + a * (c1 * b3 + c3 * b1)
       by RLVECT_1:def 3;
  E1: c1 * b1 in {c*b where c,b is Element of F : c in P & b in P} by H2,H3;
  E2: c3 * b3 in {c*b where c,b is Element of F : c in P & b in P} by H4,H5;
  a * a = (a|^1) * a by BINOM:8
       .= (a|^1) * (a|^1) by BINOM:8
       .= a|^(1+1) by BINOM:10
       .= a^2 by RING_5:3;
  then a * a in P by REALALG1:23;
  then (a * a) * (c3 * b3) in {c*b where c,b is Element of F : c in P & b in P}
      by E2,X;
  then E3: c1 * b1 + (a * a) * (c3 * b3) in
     {c1+c2 where c1,c2 is Element of F : c1 in P & c2 in P} by X,E1;
  E4: c1 * b3 in {c*b where c,b is Element of F : c in P & b in P} by H2,H5;
  c3 * b1 in {c*b where c,b is Element of F : c in P & b in P} by H4,H3;
  then c1 * b3 + c3 * b1 in
     {c1+c2 where c1,c2 is Element of F : c1 in P & c2 in P} by X,E4;
  then a * (c1 * b3 + c3 * b1) in a * P by X;
  hence o in S by H6,E3,X;
  end;
P c= S by X,P1;
then C: SQ F c= S by X;
now assume -1.F in S;
  then consider c1,c2 being Element of F such that
  H2: -1.F = c1 + c2 & c1 in P & c2 in a*P;
  consider c3 being Element of F such that H3: c2 = a * c3 & c3 in P by H2;
  0.F = (c1 + a * c3) + 1.F by H2,H3,RLVECT_1:5
     .= (c1 + 1.F) + a * c3 by RLVECT_1:def 3;
  then H4: - a * c3 = ((c1 + 1.F) + a * c3) - a * c3
                   .= (c1 + 1.F) + (a * c3 + (- a * c3)) by RLVECT_1:def 3
                   .= (c1 + 1.F) + 0.F by RLVECT_1:5;
  c3 <> 0.F by H2,H3,REALALG1:26;
  then c3" * c3 = 1.F by VECTSP_1:def 10;
  then H5: -a = (-a) * (c3 * c3")
             .= ((-a) * c3) * c3" by GROUP_1:def 3
             .= (c1 + 1.F) * c3" by H4,VECTSP_1:9;
  H: c1+1.F in {c1+c2 where c1,c2 is Element of F : c1 in P & c2 in P} by H2,X;
  c3 is non zero by H2,H3,REALALG1:26;
  then c3" in P by H3,REALALG1:27;
  then -a in {c1*c2 where c1,c2 is Element of F : c1 in P & c2 in P} by H5,H,X;
  hence contradiction by ASS,X;
  end;
then S /\ (-S) = {0.F} by C,B,REALALG1:21;
hence thesis by A,B,C,REALALG1:def 14;
end;
