
theorem lemX3a:
X^2+X+1 = X-zeta * X-(zeta^2)
proof
set F = F_Complex;
the carrier of Polynom-Ring F_Rat c= the carrier of Polynom-Ring F
   by FIELD_4:10; then
X^2+X+1 is Element of the carrier of Polynom-Ring F; then
reconsider q = X^2+X+1 as Polynomial of F;
reconsider p1 = X-zeta, p2 = X-(zeta^2) as Polynomial of F;
L: 4 -' 1 = 4 - 1 by XREAL_0:def 2; then
K: 2 -' 1 = 2 - 1 & 3 -' 1 = 3 - 1 & 3 -' 2 = 3 - 2 & 4 -' 1 = 3 &
   4 -' 2 = 4 - 2 by XREAL_0:def 2;
A: dom(p1*'p2) = NAT by FUNCT_2:def 1 .= dom q by FUNCT_2:def 1;
B: zeta|^2 = zeta^2 by lemI;
D0: q.0 = 1.F_Rat & q.1 = 1.F_Rat & q.2 = 1.F_Rat by FIELD_9:16;
    1 = 1.F by COMPLFLD:def 1,COMPLEX1:def 4; then
    -zeta - 1 = -zeta - 1.F by COMPLFLD:3; then
D2: -(-zeta - 1) = -(-zeta - 1.F) by COMPLFLD:2
                .= 1.F + --zeta by RLVECT_1:33
                .= zeta + 1.F;
now let o be object;
  assume o in dom q;
  then reconsider i = o as Element of NAT;
  consider r being FinSequence of the carrier of F such that
  B1: len r = i+1 & (p1*'p2).i = Sum r &
      for k being Element of NAT st k in dom r
      holds r.k = p1.(k-'1) * p2.(i+1-'k) by POLYNOM3:def 9;
  i <= 2 implies i = 0 or ... or i = 2; then
  per cases;
  suppose C: i = 0;
    then B2: r = <*r.1*> by B1,FINSEQ_1:40;
    then dom r = {1} by FINSEQ_1:2,FINSEQ_1:38;
    then 1 in dom r by TARSKI:def 1;
    then r.1 = p1.(1-'1) * p2.(0+1-'1) by C,B1
            .= p1.(1-'1) * p2.0 by NAT_2:8
            .= p1.0 * p2.0 by NAT_2:8
            .= rpoly(1,zeta).0 * p2.0 by FIELD_9:def 2
            .= (-power(F).(zeta,1)) * p2.0 by HURWITZ:25
            .= (-(zeta|^1)) * p2.0 by BINOM:def 2
            .= (-zeta) * p2.0 by BINOM:8
            .= (-zeta) * rpoly(1,zeta^2).0 by FIELD_9:def 2
            .= (-zeta) * (-power(F).(zeta^2,1)) by HURWITZ:25
            .= (-zeta) * (-(zeta^2|^1)) by BINOM:def 2
            .= (-zeta) * (-(zeta|^2)) by B,BINOM:8
            .= zeta * zeta|^2 by VECTSP_1:10
            .= zeta|^2 * zeta|^1 by BINOM:8
            .= zeta|^(2+1) by BINOM:10
            .= 1.F_Complex by LZ23,COMPLEX1:def 4,COMPLFLD:def 1;
    hence (p1*'p2).o = 1.F_Complex by B1,B2,RLVECT_1:44
                    .= q.o by C,D0,GAUSSINT:13,COMPLEX1:def 4,COMPLFLD:def 1;
    end;
  suppose C: i = 1;
    then B3: r = <*r.1,r.2*> by B1,FINSEQ_1:44;
    B4: dom r = {1,2} by B1,C,FINSEQ_1:def 3,FINSEQ_1:2;
    then 1 in dom r by TARSKI:def 2;
    then B5: r.1 = p1.(1-'1) * p2.(1+1-'1) by C,B1
            .= p1.0 * p2.1 by K,NAT_2:8
            .= rpoly(1,zeta).0 * p2.1 by FIELD_9:def 2
            .= (-power(F).(zeta,1)) * p2.1 by HURWITZ:25
            .= (-(zeta|^1)) * p2.1 by BINOM:def 2
            .= (-zeta) * p2.1 by BINOM:8
            .= (-zeta) * rpoly(1,zeta^2).1 by FIELD_9:def 2
            .= (-zeta) * 1_F by HURWITZ:25
            .= -zeta;
    2 in dom r by B4,TARSKI:def 2;
    then r.2 = p1.(2-'1) * p2.(1+1-'2) by C,B1
            .= p1.1 * p2.0 by K,NAT_2:8
            .= rpoly(1,zeta).1 * p2.0 by FIELD_9:def 2
            .= 1_F * p2.0 by HURWITZ:25
            .= rpoly(1,zeta^2).0 by FIELD_9:def 2
            .= -power(F).(zeta^2,1) by HURWITZ:25
            .= -(zeta^2|^1) by BINOM:def 2
            .= zeta + 1.F by D2,B,LZ23,COMPLFLD:2,BINOM:8;
    then Sum r = -zeta + (zeta + 1.F) by B3,B5,RLVECT_1:45
              .= (-zeta + zeta) + 1.F
              .= 0.F + 1.F by RLVECT_1:5;
    hence (p1*'p2).o = q.o
       by C,D0,B1,COMPLFLD:def 1,GAUSSINT:13,COMPLEX1:def 4;
    end;
  suppose C: i = 2;
    then B3: r = <*r.1,r.2,r.3*> by B1,FINSEQ_1:45;
    B4: dom r = Seg 3 by B1,C,FINSEQ_1:def 3
             .= Seg 2 \/ {2+1} by FINSEQ_1:9
             .= {1,2,3} by FINSEQ_1:2,ENUMSET1:3;
    then 1 in dom r by ENUMSET1:def 1;
    then B5: r.1 = p1.(1-'1) * p2.(2+1-'1) by C,B1
            .= p1.0 * p2.2 by K,NAT_2:8
            .= p1.0 * rpoly(1,zeta^2).2 by FIELD_9:def 2
            .= p1.0 * 0.F by HURWITZ:26;
    2 in dom r by B4,ENUMSET1:def 1;
    then B6: r.2 = p1.1 * p2.1 by K,C,B1
            .= rpoly(1,zeta).1 * p2.1 by FIELD_9:def 2
            .= 1_F * p2.1 by HURWITZ:25
            .= rpoly(1,zeta^2).1 by FIELD_9:def 2
            .= 1_F by HURWITZ:25;
    3 in dom r by B4,ENUMSET1:def 1;
    then r.3 = p1.2 * p2.(2+1-'3) by K,C,B1
            .= rpoly(1,zeta).2 * p2.(2+1-'3) by FIELD_9:def 2
            .= 0.F * p2.(2+1-'3) by HURWITZ:26;
    then Sum r = 0.F + 1.F + 0.F by B3,B5,B6,RLVECT_1:46 .= 1.F;
    hence (p1*'p2).o = q.o
       by C,D0,B1,COMPLFLD:def 1,GAUSSINT:13,COMPLEX1:def 4;
    end;
  suppose i > 2; then
    C1: i >= 2 + 1 by NAT_1:13;
    then E: q.i = 0.F_Rat by FIELD_9:16
               .= 0.F by GAUSSINT:13,COMPLFLD:def 1;
    p1 = rpoly(1,zeta) & p2 = rpoly(1,zeta^2) by FIELD_9:def 2; then
    C2: deg p1 = 1 & deg p2 = 1 by HURWITZ:27;
    C3: deg p1 = len p1 - 1 & deg p2 = len p2 - 1 by HURWITZ:def 2;
    len(p1*'p2) <= len p1 + len p2 -' 1 by leng;
    hence (p1*'p2).o = q.o by C2,C3,E,L,C1,XXREAL_0:2,ALGSEQ_1:8;
    end;
  end;
then q = p1 *' p2 by A .= X-zeta * X-(zeta^2) by POLYNOM3:def 10;
hence thesis;
end;
