
theorem XYZbS3:
for F being ordered Field,
    E being FieldExtension of F
for P being Ordering of F
for a being Element of E st a^2 in F
for f being P-quadratic non empty FinSequence of FAdj(F,{a})
ex g1,g2 being non empty FinSequence of FAdj(F,{a})
st Sum f = Sum g1 + @(FAdj(F,{a}),a) * (2 '*' Sum g2) &
   (for i being Element of NAT st i in dom g1
        ex b being non zero Element of FAdj(F,{a}),
           c1,c2 being Element of FAdj(F,{a})
        st b in P & c1 in F & c2 in F &
           g1.i = b * (c1^2 + c2^2 * @(FAdj(F,{a}),a)^2)) &
   (for i being Element of NAT st i in dom g2
        ex b being non zero Element of FAdj(F,{a}),
           c1,c2 being Element of FAdj(F,{a})
        st b in P & c1 in F & c2 in F & g2.i = b * c1 * c2)
proof
let F be ordered Field, E be FieldExtension of F;
let P be Ordering of F, a being Element of E;
assume AA: a^2 in F;
let f be P-quadratic non empty FinSequence of FAdj(F,{a});
set K = FAdj(F,{a});

defpred P[Nat] means
for f being P-quadratic non empty FinSequence of FAdj(F,{a})
st len f = $1
ex g1,g2 being non empty FinSequence of FAdj(F,{a})
st Sum f = Sum g1 + @(FAdj(F,{a}),a) * (2 '*' Sum g2) &
   (for i being Element of NAT st i in dom g1
        ex b being non zero Element of FAdj(F,{a}),
           c1,c2 being Element of FAdj(F,{a})
        st b in P & c1 in F & c2 in F &
           g1.i = b * (c1^2 + c2^2 * @(FAdj(F,{a}),a)^2)) &
   (for i being Element of NAT st i in dom g2
        ex b being non zero Element of FAdj(F,{a}),
           c1,c2 being Element of FAdj(F,{a})
        st b in P & c1 in F & c2 in F & g2.i = b * c1 * c2);

II: now let f be P-quadratic non empty FinSequence of FAdj(F,{a});
    assume len f = 1; then
    H: f = <* f.1 *> by FINSEQ_1:40; then
    dom f = { 1 } by FINSEQ_1:38,FINSEQ_1:2; then
    1 in dom f by TARSKI:def 1; then
    consider d being non zero Element of K, b being Element of K such that
    A: d in P & f.1 = d * b^2 by dq;
    consider c1,c2 being Element of K such that
    B: c1 in F & c2 in F & b = c1 + @(K,a) * c2 by AA,XYZb;
    b^2 = c1^2 + 2'*'c1 * (@(K,a) * c2) + (@(FAdj(F,{a}),a) * c2)^2
          by B,REALALG2:7
       .= c1^2 + 2'*'c1 * (@(K,a) * c2) + (@(FAdj(F,{a}),a)^2 * c2^2)
          by FIELD_9:2
       .= c1^2 + (c2^2 * @(K,a)^2) + 2'*'c1 * (@(K,a) * c2) by RLVECT_1:def 3
       .= (c1^2 + c2^2 * @(K,a)^2) + 2'*'(c1 * (c2 * @(K,a))) by REALALG2:5
       .= (c1^2 + c2^2 * @(K,a)^2) + 2'*'((c1 * c2) * @(K,a)) by GROUP_1:def 3
       .= (c1^2 + c2^2 * @(K,a)^2) + 2 '*' @(K,a) * (c1 *c2) by REALALG2:5;
    then
    E: d * b^2
       = d * (c1^2 + c2^2 * @(K,a)^2) + d * (2 '*' @(K,a) * (c1 *c2))
         by VECTSP_1:def 2
      .= d * (c1^2 + c2^2 * @(K,a)^2) + (d * (c1 *c2)) * (2 '*' @(K,a))
         by GROUP_1:def 3
      .= d * (c1^2 + c2^2 * @(K,a)^2) + (2 '*' @(K,a)) * (d * c1 * c2)
         by GROUP_1:def 3;

    set g1 = <* d * (c1^2 + c2^2 * @(FAdj(F,{a}),a)^2) *>,
        g2 = <* d * c1 * c2 *>;
    C: Sum f
          = d * (c1^2 + c2^2 * @(K,a)^2) + (2 '*' @(K,a)) * (d * c1 * c2)
            by H,A,E,RLVECT_1:44
         .= Sum g1 + (2 '*' @(K,a)) * (d * c1 * c2) by RLVECT_1:44
         .= Sum g1 + (2 '*' @(K,a)) * Sum g2 by RLVECT_1:44
         .= Sum g1 + 2 '*' (@(K,a) * Sum g2) by REALALG2:5
         .= Sum g1 + @(K,a) * (2 '*' Sum g2) by REALALG2:5;
    D: now let i be Element of NAT;
       assume D1: i in dom g1;
       dom g1 = { 1 } by FINSEQ_1:38,FINSEQ_1:2; then
       i = 1 by D1,TARSKI:def 1;
       hence ex b being non zero Element of FAdj(F,{a}),
                c1,c2 being Element of FAdj(F,{a})
          st b in P & c1 in F & c2 in F &
          g1.i = b * (c1^2 + c2^2 * @(FAdj(F,{a}),a)^2) by A,B;
       end;
    now let i be Element of NAT;
      assume D1: i in dom g2;
      dom g2 = { 1 } by FINSEQ_1:38,FINSEQ_1:2; then
      i = 1 by D1,TARSKI:def 1;
      hence ex b being non zero Element of FAdj(F,{a}),
            c1,c2 being Element of FAdj(F,{a})
      st b in P & c1 in F & c2 in F & g2.i = b * c1 * c2 by A,B;
      end;
    hence ex g1,g2 being non empty FinSequence of FAdj(F,{a})
       st Sum f = Sum g1 + @(FAdj(F,{a}),a) * (2 '*' Sum g2) &
          (for i being Element of NAT st i in dom g1
          ex b being non zero Element of FAdj(F,{a}),
             c1,c2 being Element of FAdj(F,{a})
          st b in P & c1 in F & c2 in F &
             g1.i = b * (c1^2 + c2^2 * @(FAdj(F,{a}),a)^2)) &
         (for i being Element of NAT st i in dom g2
          ex b being non zero Element of FAdj(F,{a}),
             c1,c2 being Element of FAdj(F,{a})
          st b in P & c1 in F & c2 in F & g2.i = b * c1 * c2) by C,D;
      end; then
IA: P[1];
IS: now let k be non zero Nat;
    assume IV: P[k];
    now let f be P-quadratic non empty FinSequence of FAdj(F,{a});
      assume AS: len f = k + 1;
      consider G being FinSequence, y being object such that
      A1: f = G^<*y*> by FINSEQ_1:46;
      rng G c= rng f by A1,FINSEQ_1:29; then
      reconsider G as FinSequence of K by XBOOLE_1:1,FINSEQ_1:def 4;
      A5: rng f c= the carrier of K;
      A6: rng<*y*> c= rng f by A1,FINSEQ_1:30;
      rng<*y*> = {y} & y in {y} by FINSEQ_1:38,TARSKI:def 1; then
      reconsider y as Element of K by A5,A6;
      A3: len f = len G + len<*y*> by A1,FINSEQ_1:22
               .= len G + 1 by FINSEQ_1:39;
      per cases;

      suppose G is empty;
      then f = <*y*> by A1,FINSEQ_1:34;
      then len f = 1 by FINSEQ_1:40;
      hence ex g1,g2 being non empty FinSequence of FAdj(F,{a})
         st Sum f = Sum g1 + @(FAdj(F,{a}),a) * (2 '*' Sum g2) &
         (for i being Element of NAT st i in dom g1
          ex b being non zero Element of FAdj(F,{a}),
             c1,c2 being Element of FAdj(F,{a})
          st b in P & c1 in F & c2 in F &
             g1.i = b * (c1^2 + c2^2 * @(FAdj(F,{a}),a)^2)) &
         (for i being Element of NAT st i in dom g2
          ex b being non zero Element of FAdj(F,{a}),
             c1,c2 being Element of FAdj(F,{a})
          st b in P & c1 in F & c2 in F & g2.i = b * c1 * c2) by II;
      end;

      suppose A5: G is non empty;
      <*y*> is FinSequence of K & f = G^<*y*> by A1; then
      B: G is P-quadratic & <*y*> is P-quadratic by XYZbS3a; then
      consider h1,h2 being non empty FinSequence of K such that
      C: Sum G = Sum h1 + @(K,a) * (2 '*' Sum h2) &
        (for i being Element of NAT st i in dom h1
         ex b being non zero Element of K, c1,c2 being Element of K
         st b in P & c1 in F & c2 in F &
            h1.i = b * (c1^2 + c2^2 * @(K,a)^2)) &
        (for i being Element of NAT st i in dom h2
         ex b being non zero Element of K, c1,c2 being Element of K
         st b in P & c1 in F & c2 in F & h2.i = b * c1 * c2) by A5,AS,A3,IV;

      len <*y*> = 1 by FINSEQ_1:40; then
      consider y1,y2 being non empty FinSequence of K such that
      D: Sum <*y*> = Sum y1 + @(K,a) * (2 '*' Sum y2) &
        (for i being Element of NAT st i in dom y1
         ex b being non zero Element of K, c1,c2 being Element of K
         st b in P & c1 in F & c2 in F &
            y1.i = b * (c1^2 + c2^2 * @(K,a)^2)) &
        (for i being Element of NAT st i in dom y2
         ex b being non zero Element of K, c1,c2 being Element of K
         st b in P & c1 in F & c2 in F & y2.i = b * c1 * c2) by B,II;

      set g1 = h1^y1, g2 = h2^y2;
      E: Sum g1 + @(K,a) * (2 '*' Sum g2)
       = (Sum h1 + Sum y1) + @(K,a) * (2 '*' Sum g2) by RLVECT_1:41
      .= (Sum h1 + Sum y1) + @(K,a) * (2 '*' (Sum h2 + Sum y2)) by RLVECT_1:41
      .= (Sum h1 + Sum y1) + 2 '*' ((Sum h2 + Sum y2) * @(K,a)) by REALALG2:5
      .= (Sum h1 + Sum y1) + 2 '*' (Sum h2 * @(K,a) + Sum y2 * @(K,a))
         by VECTSP_1:def 3
      .= (Sum h1 + Sum y1) + ((Sum h2 * @(K,a) + Sum y2 * @(K,a)) +
                              (Sum h2 * @(K,a) + Sum y2 * @(K,a))) by RING_5:2
      .= (Sum h1 + Sum y1) + (Sum h2 * @(K,a) + (Sum y2 * @(K,a) +
                              (Sum h2 * @(K,a) + Sum y2 * @(K,a))))
         by RLVECT_1:def 3
      .= (Sum h1 + Sum y1) + (Sum h2 * @(K,a) + ((Sum y2 * @(K,a) +
                              Sum y2 * @(K,a)) + Sum h2 * @(K,a)))
         by RLVECT_1:def 3
      .= (Sum h1 + Sum y1) + ((Sum h2 * @(K,a) + Sum h2 * @(K,a)) +
                              (Sum y2 * @(K,a) + Sum y2 * @(K,a)))
         by RLVECT_1:def 3
      .= (Sum h1 + Sum y1) + ((2 '*' (Sum h2 * @(K,a))) +
                              (Sum y2 * @(K,a) + Sum y2 * @(K,a))) by RING_5:2
      .= (Sum h1 + Sum y1) + ((2 '*' Sum h2 * @(K,a)) +
                              (Sum y2 * @(K,a) + Sum y2 * @(K,a)))
         by REALALG2:5
      .= (Sum h1 + Sum y1) + ((2'*'Sum h2 * @(K,a)) + 2'*'(Sum y2 * @(K,a)))
         by RING_5:2
      .= (Sum h1 + Sum y1) + (2 '*' Sum h2 * @(K,a) + 2 '*' Sum y2 * @(K,a))
         by REALALG2:5
      .= ((Sum h1 + Sum y1) + 2 '*' Sum h2 * @(K,a)) + 2 '*' Sum y2 * @(K,a)
         by RLVECT_1:def 3
      .= (Sum y1 + (Sum h1 + 2 '*' Sum h2 * @(K,a))) + 2 '*' Sum y2 * @(K,a)
         by RLVECT_1:def 3
      .= (Sum h1 + 2 '*' Sum h2 * @(K,a)) + (Sum y1 + 2 '*' Sum y2 * @(K,a))
         by RLVECT_1:def 3
      .= Sum f by C,D,A1,RLVECT_1:41;

      F: now let i be Element of NAT;
         assume i in dom g1; then
         per cases by FINSEQ_1:25;
         suppose F1: i in dom h1;
           then g1.i = h1.i by FINSEQ_1:def 7;
           hence ex b being non zero Element of FAdj(F,{a}),
                    c1,c2 being Element of FAdj(F,{a})
                 st b in P & c1 in F & c2 in F &
                    g1.i = b * (c1^2 + c2^2 * @(FAdj(F,{a}),a)^2) by F1,C;
           end;
         suppose ex n being Nat st n in dom y1 & i = len h1 + n; then
           consider n being Nat such that F1: n in dom y1 & i = len h1 + n;
           ex b being non zero Element of FAdj(F,{a}),
              c1,c2 being Element of FAdj(F,{a})
              st b in P & c1 in F & c2 in F &
                 y1.n = b * (c1^2 + c2^2 * @(FAdj(F,{a}),a)^2) by F1,D;
           hence ex b being non zero Element of FAdj(F,{a}),
                    c1,c2 being Element of FAdj(F,{a})
                 st b in P & c1 in F & c2 in F &
                    g1.i = b * (c1^2 + c2^2 * @(K,a)^2) by F1,FINSEQ_1:def 7;
           end;
         end;

        now let i be Element of NAT;
         assume i in dom g2; then
         per cases by FINSEQ_1:25;
         suppose F1: i in dom h2;
           then g2.i = h2.i by FINSEQ_1:def 7;
           hence ex b being non zero Element of FAdj(F,{a}),
                    c1,c2 being Element of FAdj(F,{a})
                 st b in P & c1 in F & c2 in F & g2.i = b * c1 * c2 by F1,C;
           end;
         suppose ex n being Nat st n in dom y2 & i = len h2 + n; then
           consider n being Nat such that F1: n in dom y2 & i = len h2 + n;
           ex b being non zero Element of FAdj(F,{a}),
              c1,c2 being Element of FAdj(F,{a})
              st b in P & c1 in F & c2 in F & y2.n = b * c1 * c2 by F1,D;
           hence ex b being non zero Element of FAdj(F,{a}),
                    c1,c2 being Element of FAdj(F,{a})
                 st b in P & c1 in F & c2 in F & g2.i = b * c1 * c2
              by F1,FINSEQ_1:def 7;
           end;
         end;

      hence ex g1,g2 being non empty FinSequence of FAdj(F,{a})
         st Sum f = Sum g1 + @(FAdj(F,{a}),a) * (2 '*' Sum g2) &
         (for i being Element of NAT st i in dom g1
          ex b being non zero Element of FAdj(F,{a}),
             c1,c2 being Element of FAdj(F,{a})
          st b in P & c1 in F & c2 in F &
             g1.i = b * (c1^2 + c2^2 * @(FAdj(F,{a}),a)^2)) &
         (for i being Element of NAT st i in dom g2
          ex b being non zero Element of FAdj(F,{a}),
             c1,c2 being Element of FAdj(F,{a})
          st b in P & c1 in F & c2 in F & g2.i = b * c1 * c2) by E,F;
      end;
      end;
    hence P[k+1];
    end;
I: for k being non zero Nat holds P[k] from NAT_1:sch 10(IA,IS);
consider n being Nat such that H: n = len f;
thus thesis by H,I;
end;
