reserve x for set;
reserve i,j for Integer;
reserve n,n1,n2,n3 for Nat;
reserve p for Prime;
reserve a,b,c,d for Element of GF(p);
reserve K for Ring;
reserve a1,a2,a3,a4,a5,a6 for Element of K;
reserve px,py,pz for object;
reserve Px,Py,Pz for Element of GF(p);
reserve P for Element of ProjCo(GF(p));
reserve O for Element of EC_SetProjCo(a,b,p);

theorem Th57:
  for p be 5_or_greater Prime, z be Element of EC_WParam p,
  g2, gf1, gf2, gf3 be Element of GF(p),
  P, Q be Element of EC_SetProjCo(z`1,z`2,p),
  R be Element of
  [:the carrier of GF(p), the carrier of GF(p), the carrier of GF(p):]
  st g2 = 2 mod p & gf1 = Q`2_3*(P`3_3) - P`2_3*(Q`3_3) &
    gf2 = Q`1_3*(P`3_3) - P`1_3*(Q`3_3) &
  gf3 = (gf1 |^2)*(P`3_3)*(Q`3_3) - (gf2 |^3) - g2*(gf2 |^2)*(P`1_3)*(Q`3_3) &
  R = [gf2*gf3,
     gf1 * ((gf2 |^2)*(P`1_3)*(Q`3_3)-gf3) - (gf2 |^3)*(P`2_3)*(Q`3_3),
  (gf2 |^3)*(P`3_3)*(Q`3_3)]
  holds z`1*(gf2 |^2)*(P`3_3)*(Q`3_3)*(R`3_3) =
  (gf2 |^2)*(P`1_3*(Q`1_3)*(R`3_3)+P`3_3*(Q`1_3)*(R`1_3)+P`1_3*(Q`3_3)*(R`1_3))
  + g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3) - gf1*(P`1_3))
  proof
    let p be 5_or_greater Prime, z be Element of EC_WParam p,
    g2, gf1, gf2, gf3 be Element of GF(p),
    P, Q be Element of EC_SetProjCo(z`1,z`2,p),
    R be Element of
    [:the carrier of GF(p), the carrier of GF(p), the carrier of GF(p):]
    such that
    A1: g2 = 2 mod p and
    A2: gf1 = Q`2_3*(P`3_3) - P`2_3*(Q`3_3) &
          gf2 = Q`1_3*(P`3_3) - P`1_3*(Q`3_3) &
    gf3 = (gf1 |^2)*(P`3_3)*(Q`3_3)
         - (gf2 |^3) - g2*(gf2 |^2)*(P`1_3)*(Q`3_3)  and
A3: R = [gf2*gf3, gf1 * ((gf2 |^2)*(P`1_3)*(Q`3_3)-gf3)
             - (gf2 |^3)*(P`2_3)*(Q`3_3),
    (gf2 |^3)*(P`3_3)*(Q`3_3)];
    set a = z`1;
    set b = z`2;
    A4: gf2*(P`2_3)-gf1*(P`1_3)
    = ((Q`1_3*(P`3_3))*(P`2_3)-(P`1_3*(Q`3_3))*(P`2_3))
          -(Q`2_3*(P`3_3)-P`2_3*(Q`3_3))*(P`1_3) by A2,VECTSP_1:11
    .= ((Q`1_3*(P`3_3))*(P`2_3)-(P`1_3*(Q`3_3))*(P`2_3))
            -((Q`2_3*(P`3_3))*(P`1_3)-(P`2_3*(Q`3_3))*(P`1_3))
    by VECTSP_1:11
    .= ((Q`1_3*(P`3_3))*(P`2_3)-(P`1_3*(Q`3_3))*(P`2_3))
     +((P`2_3*(Q`3_3))*(P`1_3)-(Q`2_3*(P`3_3))*(P`1_3))
    by VECTSP_1:17
    .= ((Q`1_3*(P`3_3))*(P`2_3)-P`1_3*(Q`3_3*(P`2_3)))
     +((P`2_3*(Q`3_3))*(P`1_3)-(Q`2_3*(P`3_3))*(P`1_3))
    by GROUP_1:def 3
    .= (Q`1_3*(P`3_3))*(P`2_3)+(-P`1_3*(Q`3_3*(P`2_3))
    +((P`2_3*(Q`3_3))*(P`1_3)-(Q`2_3*(P`3_3))*(P`1_3)))
    by ALGSTR_1:7
    .= (Q`1_3*(P`3_3))*(P`2_3)+((-P`1_3*(P`2_3*(Q`3_3))
    +(P`2_3*(Q`3_3))*(P`1_3))-(Q`2_3*(P`3_3))*(P`1_3))
    by ALGSTR_1:7
    .= (Q`1_3*(P`3_3))*(P`2_3)+(0.GF(p)-(P`3_3*(Q`2_3))*(P`1_3)) by VECTSP_1:19
    .= (P`3_3*(Q`1_3))*(P`2_3)-(P`3_3*(Q`2_3))*(P`1_3) by VECTSP_1:18
    .= P`3_3*(Q`1_3*(P`2_3))-(P`3_3*(Q`2_3))*(P`1_3) by GROUP_1:def 3
    .= P`3_3*(P`2_3*(Q`1_3))-P`3_3*(Q`2_3*(P`1_3)) by GROUP_1:def 3
    .= P`3_3*(P`2_3*(Q`1_3)-Q`2_3*(P`1_3)) by VECTSP_1:11;
    gf2*(Q`2_3)-gf1*(Q`1_3)
    = ((Q`1_3*(P`3_3))*(Q`2_3)-(P`1_3*(Q`3_3))*(Q`2_3))
    -(Q`2_3*(P`3_3)-P`2_3*(Q`3_3))*(Q`1_3) by A2,VECTSP_1:11
    .= ((Q`1_3*(P`3_3))*(Q`2_3)-(P`1_3*(Q`3_3))*(Q`2_3))
    -((Q`2_3*(P`3_3))*(Q`1_3)-(P`2_3*(Q`3_3))*(Q`1_3))
    by VECTSP_1:11
    .= ((Q`1_3*(P`3_3))*(Q`2_3)-(P`1_3*(Q`3_3))*(Q`2_3))
    +((P`2_3*(Q`3_3))*(Q`1_3)-(Q`2_3*(P`3_3))*(Q`1_3))
    by VECTSP_1:17
    .= (Q`1_3*(P`3_3))*(Q`2_3)-(P`1_3*(Q`3_3))*(Q`2_3)
    +(P`2_3*(Q`3_3))*(Q`1_3)-(Q`2_3*(P`3_3))*(Q`1_3)
    by ALGSTR_1:7
    .= (Q`1_3*(P`3_3))*(Q`2_3)-(Q`2_3*(P`3_3))*(Q`1_3)
    +(P`2_3*(Q`3_3))*(Q`1_3)-(P`1_3*(Q`3_3))*(Q`2_3)
    by Th7
    .= Q`1_3*(P`3_3*(Q`2_3))-Q`1_3*(P`3_3*(Q`2_3))
    +(Q`3_3*(P`2_3))*(Q`1_3)-(Q`3_3*(P`1_3))*(Q`2_3)
    by GROUP_1:def 3
    .= 0.GF(p)+(Q`3_3*(P`2_3))*(Q`1_3)-(Q`3_3*(P`1_3))*(Q`2_3) by VECTSP_1:19
    .= (Q`3_3*(P`2_3))*(Q`1_3)-(Q`3_3*(P`1_3))*(Q`2_3) by ALGSTR_1:7
    .= Q`3_3*(P`2_3*(Q`1_3))-(Q`3_3*(P`1_3))*(Q`2_3) by GROUP_1:def 3
    .= Q`3_3*(P`2_3*(Q`1_3))-Q`3_3*(P`1_3*(Q`2_3)) by GROUP_1:def 3
    .= Q`3_3*(P`2_3*(Q`1_3)-P`1_3*(Q`2_3)) by VECTSP_1:11; then
    A5: (gf2*(Q`2_3)-gf1*(Q`1_3))*(P`3_3)
    = (gf2*(P`2_3)-gf1*(P`1_3))*(Q`3_3) by A4,GROUP_1:def 3;
    A6: P`1_3*(Q`1_3)*(R`3_3)+P`3_3*(Q`1_3)*(R`1_3)+P`1_3*(Q`3_3)*(R`1_3)
    = P`1_3*(Q`1_3)*(R`3_3)+(P`3_3*(Q`1_3)*(R`1_3)+P`1_3*(Q`3_3)*(R`1_3))
       by ALGSTR_1:7
    .= P`1_3*(Q`1_3)*(R`3_3)+(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`1_3)
          by VECTSP_1:def 7
    .= P`1_3*(Q`1_3)*((gf2 |^3)*(P`3_3*(Q`3_3)))
    +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(gf2*gf3) by GROUP_1:def 3,A3
    .= ((P`1_3*(Q`1_3))*(gf2 |^(2+1)))*(P`3_3*(Q`3_3))
    +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(gf2*gf3) by GROUP_1:def 3
    .= ((P`1_3*(Q`1_3))*((gf2 |^2)*gf2))*(P`3_3*(Q`3_3))
    +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(gf2*gf3) by EC_PF_1:24
    .= ((P`1_3*(Q`1_3))*((gf2 |^2)*gf2))*(P`3_3*(Q`3_3))
    +((P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*gf2)*gf3 by GROUP_1:def 3
    .= (((P`1_3*(Q`1_3))*(gf2 |^2))*gf2)*(P`3_3*(Q`3_3))
    +(gf2*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3)))*gf3 by GROUP_1:def 3
    .= (gf2*((P`1_3*(Q`1_3))*(gf2 |^2)))*(P`3_3*(Q`3_3))
    +gf2*((P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*gf3) by GROUP_1:def 3
    .= gf2*(((P`1_3*(Q`1_3))*(gf2 |^2))*(P`3_3*(Q`3_3)))
    +gf2*((P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*gf3) by GROUP_1:def 3
    .= gf2*(((P`1_3*(Q`1_3))*(gf2 |^2))*(P`3_3*(Q`3_3))
    +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*gf3) by VECTSP_1:def 7;
A7: gf3 = (gf1 |^2)*(P`3_3)*(Q`3_3)-(gf2 |^(2+1))-g2*(gf2 |^2)*(P`1_3)*(Q`3_3)
          by A2
    .= (gf1 |^2)*(P`3_3)*(Q`3_3)-(gf2 |^2)*gf2-g2*(gf2 |^2)*(P`1_3)*(Q`3_3)
            by EC_PF_1:24
    .= (gf1 |^2)*(P`3_3)*(Q`3_3)+(-(gf2 |^2)*gf2-g2*(gf2 |^2)*(P`1_3)*(Q`3_3))
               by ALGSTR_1:7
    .= (gf1 |^2)*(P`3_3)*(Q`3_3)+(-(gf2 |^2)*gf2
           -(g2*(gf2 |^2))*(P`1_3*(Q`3_3)))
    by GROUP_1:def 3
    .= (gf1 |^2)*(P`3_3)*(Q`3_3)
           +(-(gf2 |^2)*gf2-(gf2 |^2)*(g2*(P`1_3*(Q`3_3))))
    by GROUP_1:def 3
    .= (gf1 |^2)*(P`3_3)*(Q`3_3)+((-(gf2 |^2))*gf2-(gf2 |^2)
    *(g2*(P`1_3*(Q`3_3)))) by VECTSP_1:9
    .= (gf1 |^2)*(P`3_3)*(Q`3_3)+((-(gf2 |^2))*gf2+(-(gf2 |^2))
    *(g2*(P`1_3*(Q`3_3)))) by VECTSP_1:9
    .= (gf1 |^2)*(P`3_3)*(Q`3_3)+(-(gf2 |^2))*((Q`1_3*(P`3_3)-P`1_3*(Q`3_3))
    +g2*(P`1_3*(Q`3_3))) by A2,VECTSP_1:def 7
    .= (gf1 |^2)*(P`3_3)*(Q`3_3)-(gf2 |^2)*((Q`1_3*(P`3_3)
     -P`1_3*(Q`3_3))+g2*(P`1_3*(Q`3_3)))
    by VECTSP_1:9
    .= (gf1 |^2)*(P`3_3)*(Q`3_3)-(gf2 |^2)*(Q`1_3*(P`3_3)
    +(-P`1_3*(Q`3_3)+g2*(P`1_3*(Q`3_3))))
    by ALGSTR_1:7
    .= (gf1 |^2)*(P`3_3)*(Q`3_3)-(gf2 |^2)*(Q`1_3*(P`3_3)
     +(g2*(P`1_3*(Q`3_3))-P`1_3*(Q`3_3)))
    .= (gf1 |^2)*(P`3_3)*(Q`3_3)-(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))
          by A1,Th24;
    ((P`1_3*(Q`1_3))*(gf2 |^2))*(P`3_3*(Q`3_3))
        +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*gf3
    = (gf2 |^2)*((P`1_3*(Q`1_3))*(P`3_3*(Q`3_3)))
          +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*gf3
    by GROUP_1:def 3
    .= (gf2 |^2)*(P`3_3*(Q`3_3*(P`1_3*(Q`1_3))))
             +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*gf3
    by GROUP_1:def 3
    .= (gf2 |^2)*(P`3_3*((Q`3_3*(P`1_3))*(Q`1_3)))
              +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*gf3
    by GROUP_1:def 3
    .= (gf2 |^2)*((P`3_3*(Q`1_3))*(P`1_3*(Q`3_3)))
            +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*gf3
    by GROUP_1:def 3
    .= (-(gf2 |^2))*(-(P`3_3*(Q`1_3))*(P`1_3*(Q`3_3)))
    +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3)
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))) by A7,VECTSP_1:10
    .= (-(gf2 |^2))*(-(P`3_3*(Q`1_3))*(P`1_3*(Q`3_3)))
    +((P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3))
    -(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))))
       by VECTSP_1:11
    .= (-(gf2 |^2))*(-(P`3_3*(Q`1_3))*(P`1_3*(Q`3_3)))
    +(-(gf2 |^2)*((P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3)))
    +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3)))
          by GROUP_1:def 3
    .= (-(gf2 |^2))*(-(P`3_3*(Q`1_3))*(P`1_3*(Q`3_3)))
    +(-(gf2 |^2)*((P`3_3*(Q`1_3)+P`1_3*(Q`3_3)) |^2)
    +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3))) by EC_PF_1:22
    .= (-(gf2 |^2))*(-(P`3_3*(Q`1_3))*(P`1_3*(Q`3_3)))
    +((-(gf2 |^2))*((P`3_3*(Q`1_3)+P`1_3*(Q`3_3)) |^2)
    +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3))) by VECTSP_1:9
    .= ((-(gf2 |^2))*(-(P`3_3*(Q`1_3))*(P`1_3*(Q`3_3)))
    +(-(gf2 |^2))*((P`3_3*(Q`1_3)+P`1_3*(Q`3_3)) |^2))
    +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3)) by ALGSTR_1:7
    .= (-(gf2 |^2))*((-(P`3_3*(Q`1_3))*(P`1_3*(Q`3_3)))
    +((P`3_3*(Q`1_3)+P`1_3*(Q`3_3)) |^2))
    +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3)) by
VECTSP_1:def 7
    .= (-(gf2 |^2))*((((P`3_3*(Q`1_3)) |^2 +g2*(P`3_3*(Q`1_3))*(P`1_3*(Q`3_3))
    +(P`1_3*(Q`3_3)) |^2)) -(P`3_3*(Q`1_3))*(P`1_3*(Q`3_3)))
    +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3)) by A1,Th25
    .= (-(gf2 |^2))*(((P`3_3*(Q`1_3)) |^2)+g2*(P`3_3*(Q`1_3))*(P`1_3*(Q`3_3))
    -(P`3_3*(Q`1_3))*(P`1_3*(Q`3_3))+((P`1_3*(Q`3_3)) |^2))
    +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3)) by ALGSTR_1:8
    .= (-(gf2 |^2))*(((P`3_3*(Q`1_3)) |^2)+g2*((P`3_3*(Q`1_3))*(P`1_3*(Q`3_3)))
    -(P`3_3*(Q`1_3))*(P`1_3*(Q`3_3))+((P`1_3*(Q`3_3)) |^2))
    +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3)) by GROUP_1:def 3
    .= (-(gf2 |^2))*(((P`3_3*(Q`1_3)) |^2)
          +(g2*((P`3_3*(Q`1_3))*(P`1_3*(Q`3_3)))
    -(P`3_3*(Q`1_3))*(P`1_3*(Q`3_3)))+((P`1_3*(Q`3_3)) |^2))
    +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3)) by ALGSTR_1:7
    .= (-(gf2 |^2))*(((P`3_3*(Q`1_3)) |^2)+(P`3_3*(Q`1_3))*(P`1_3*(Q`3_3))
    +((P`1_3*(Q`3_3)) |^2))
    +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3)) by A1,Th24
    .= (-gf2*gf2)*(((P`3_3*(Q`1_3)) |^2)+(P`3_3*(Q`1_3))*(P`1_3*(Q`3_3))
    +((P`1_3*(Q`3_3)) |^2))
    +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3)) by EC_PF_1:22
    .= ((-gf2)*gf2)*(((P`3_3*(Q`1_3)) |^2)+(P`3_3*(Q`1_3))*(P`1_3*(Q`3_3))
    +((P`1_3*(Q`3_3)) |^2))
    +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3)) by VECTSP_1:9
    .= (-gf2)*((Q`1_3*(P`3_3)-P`1_3*(Q`3_3))*
    (((P`3_3*(Q`1_3)) |^2)+(P`3_3*(Q`1_3))*(P`1_3*(Q`3_3))
           +((P`1_3*(Q`3_3)) |^2)))
    +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3))
   by A2,GROUP_1:def 3
    .= (-gf2)*(((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3))
    +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3)) by Th17; then
A8: (gf2 |^2)*(P`1_3*(Q`1_3)*(R`3_3)
         +P`3_3*(Q`1_3)*(R`1_3)+P`1_3*(Q`3_3)*(R`1_3))
    = ((gf2 |^2)*gf2)*((-gf2)*(((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3))
    +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3)))
    by A6,GROUP_1:def 3
    .= (gf2 |^(2+1))*((-gf2)*(((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3))
    +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3))) by EC_PF_1:24
    .= (gf2 |^3)*((-gf2)*(((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3))
    +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3)));
    ((P`2_3) |^2)*(P`3_3) - ((P`1_3) |^3 +
       a*(P`1_3)*(P`3_3) |^2 + b*(P`3_3) |^3)
    = 0.GF(p) by Th35; then
    (Q`3_3 |^3)*((P`2_3 |^2)*(P`3_3))
    = (Q`3_3 |^3)*((P`1_3 |^3)+a*(P`1_3)*(P`3_3 |^2)+b*(P`3_3 |^3))
      by VECTSP_1:19
    .= (Q`3_3 |^3)*((P`1_3 |^3)+a*(P`1_3)*(P`3_3 |^2))
    +(Q`3_3 |^3)*(b*(P`3_3 |^3)) by VECTSP_1:def 7
    .= (Q`3_3 |^3)*(P`1_3 |^3)+(Q`3_3 |^3)*(a*(P`1_3)*(P`3_3 |^2))
    +(Q`3_3 |^3)*(b*(P`3_3 |^3)) by VECTSP_1:def 7
    .= ((Q`3_3*(P`1_3)) |^3)+(a*(P`1_3)*(P`3_3 |^2))*(Q`3_3 |^(2+1))
    +(b*(P`3_3 |^3))*(Q`3_3 |^3) by BINOM:9
    .= ((P`1_3*(Q`3_3)) |^3)+(a*(P`1_3)*(P`3_3 |^2))*((Q`3_3 |^2)*(Q`3_3))
    +b*(P`3_3 |^3)*(Q`3_3 |^3) by EC_PF_1:24
    .= ((P`1_3*(Q`3_3)) |^3)+(a*(P`1_3))*((P`3_3 |^2)*((Q`3_3 |^2)*(Q`3_3)))
    +b*(P`3_3 |^3)*(Q`3_3 |^3) by GROUP_1:def 3
    .= ((P`1_3*(Q`3_3)) |^3)+(a*(P`1_3))*(((P`3_3 |^2)*(Q`3_3 |^2))*(Q`3_3))
    +b*(P`3_3 |^3)*(Q`3_3 |^3) by GROUP_1:def 3
    .= ((P`1_3*(Q`3_3)) |^3)+(a*(P`1_3))*(((P`3_3*(Q`3_3)) |^2)*(Q`3_3))
    +b*(P`3_3 |^3)*(Q`3_3 |^3) by BINOM:9
    .= ((P`1_3*(Q`3_3)) |^3)+a*(P`1_3*(((P`3_3*(Q`3_3)) |^2)*(Q`3_3)))
    +b*(P`3_3 |^3)*(Q`3_3 |^3) by GROUP_1:def 3
    .= ((P`1_3*(Q`3_3)) |^3)+a*(((P`3_3*(Q`3_3)) |^2)*(Q`3_3*(P`1_3)))
    +b*(P`3_3 |^3)*(Q`3_3 |^3) by GROUP_1:def 3
    .= ((P`1_3*(Q`3_3)) |^3)+(a*((P`3_3*(Q`3_3)) |^2))*(P`1_3*(Q`3_3))
    +b*(P`3_3 |^3)*(Q`3_3 |^3) by GROUP_1:def 3; then
    A9: -(Q`3_3 |^3)*((P`2_3 |^2)*(P`3_3))
    = -b*(P`3_3 |^3)*(Q`3_3 |^3)-(((P`1_3*(Q`3_3)) |^3)
    +(a*((P`3_3*(Q`3_3)) |^2))*(P`1_3*(Q`3_3))) by VECTSP_1:17
    .= -b*(P`3_3 |^3)*(Q`3_3 |^3)+(-(a*((P`3_3*(Q`3_3)) |^2))*(P`1_3*(Q`3_3))
    -((P`1_3*(Q`3_3)) |^3)) by VECTSP_1:17;
    ((Q`2_3) |^2)*(Q`3_3) - ((Q`1_3) |^3
              + a*(Q`1_3)*(Q`3_3) |^2 + b*(Q`3_3) |^3)
    = 0.GF(p) by Th35; then
    A10: (P`3_3 |^3)*(((Q`2_3) |^2)*(Q`3_3))
    = (P`3_3 |^3)*((Q`1_3 |^3)+a*(Q`1_3)*(Q`3_3 |^2)+b*(Q`3_3 |^3))
           by VECTSP_1:19
    .= (P`3_3 |^3)*((Q`1_3 |^3)+a*(Q`1_3)*(Q`3_3 |^2))
    +(P`3_3 |^3)*(b*(Q`3_3 |^3)) by VECTSP_1:def 7
    .= (P`3_3 |^3)*(Q`1_3 |^3)+(P`3_3 |^3)*(a*(Q`1_3)*(Q`3_3 |^2))
    +(P`3_3 |^3)*(b*(Q`3_3 |^3)) by VECTSP_1:def 7
    .= ((P`3_3*(Q`1_3)) |^3)+(a*(Q`1_3)*(Q`3_3 |^2))*(P`3_3 |^(2+1))
    +(b*(Q`3_3 |^3))*(P`3_3 |^3) by BINOM:9
    .= ((P`3_3*(Q`1_3)) |^3)+(a*(Q`1_3)*(Q`3_3 |^2))*((P`3_3 |^2)*(P`3_3))
    +(b*((Q`3_3 |^3))*(P`3_3 |^3)) by EC_PF_1:24
    .= ((P`3_3*(Q`1_3)) |^3)+(a*(Q`1_3)*(Q`3_3 |^2))*((P`3_3 |^2)*(P`3_3))
    +b*((Q`3_3 |^3)*(P`3_3 |^3)) by GROUP_1:def 3
    .= ((P`3_3*(Q`1_3)) |^3)+(a*(Q`1_3))*((Q`3_3 |^2)*((P`3_3 |^2)*(P`3_3)))
    +b*((P`3_3 |^3)*(Q`3_3 |^3)) by GROUP_1:def 3
    .= ((P`3_3*(Q`1_3)) |^3)+(a*(Q`1_3))*(((Q`3_3 |^2)*(P`3_3 |^2))*(P`3_3))
    +b*((P`3_3 |^3)*(Q`3_3 |^3)) by GROUP_1:def 3
    .= ((P`3_3*(Q`1_3)) |^3)+(a*(Q`1_3))*(((Q`3_3*(P`3_3)) |^2)*(P`3_3))
    +b*((P`3_3 |^3)*(Q`3_3 |^3)) by BINOM:9
    .= ((P`3_3*(Q`1_3)) |^3)+(a*(Q`1_3))*(((P`3_3*(Q`3_3)) |^2)*(P`3_3))
    +b*(P`3_3 |^3)*(Q`3_3 |^3) by GROUP_1:def 3
    .= ((P`3_3*(Q`1_3)) |^3)+a*(Q`1_3*(((P`3_3*(Q`3_3)) |^2)*(P`3_3)))
    +b*(P`3_3 |^3)*(Q`3_3 |^3) by GROUP_1:def 3
    .= ((P`3_3*(Q`1_3)) |^3)+a*(((P`3_3*(Q`3_3)) |^2)*(P`3_3*(Q`1_3)))
    +b*(P`3_3 |^3)*(Q`3_3 |^3) by GROUP_1:def 3
    .= ((P`3_3*(Q`1_3)) |^3)+(a*((P`3_3*(Q`3_3)) |^2))*(Q`1_3*(P`3_3))
    +b*(P`3_3 |^3)*(Q`3_3 |^3) by GROUP_1:def 3;
    A11: (gf1*(P`3_3*(Q`3_3)))*(P`2_3*(Q`3_3)+Q`2_3*(P`3_3))
    = (P`3_3*(Q`3_3))*((Q`2_3*(P`3_3)
      -P`2_3*(Q`3_3))*(Q`2_3*(P`3_3)+P`2_3*(Q`3_3)))
         by A2,GROUP_1:def 3
    .= (P`3_3*(Q`3_3))*(((Q`2_3*(P`3_3)) |^2)-((P`2_3*(Q`3_3)) |^2)) by Th15
    .= (P`3_3*(Q`3_3))*((Q`2_3 |^2)*(P`3_3 |^2)-((P`2_3*(Q`3_3)) |^2))
     by BINOM:9
    .= (P`3_3*(Q`3_3))*((Q`2_3 |^2)*(P`3_3 |^2)-(P`2_3 |^2)*(Q`3_3 |^2))
           by BINOM:9
    .= (P`3_3*(Q`3_3))*((Q`2_3 |^2)*(P`3_3 |^2))
    -(P`3_3*(Q`3_3))*((P`2_3 |^2)*(Q`3_3 |^2)) by VECTSP_1:11
    .= P`3_3*(Q`3_3*((Q`2_3 |^2)*(P`3_3 |^2)))
    -(P`3_3*(Q`3_3))*((Q`3_3 |^2)*(P`2_3 |^2)) by GROUP_1:def 3
    .= P`3_3*((Q`3_3*(Q`2_3 |^2))*(P`3_3 |^2))
    -(P`3_3*(Q`3_3))*((Q`3_3 |^2)*(P`2_3 |^2)) by GROUP_1:def 3
    .= P`3_3*((P`3_3 |^2)*(Q`3_3*(Q`2_3 |^2)))
    -P`3_3*(Q`3_3*((Q`3_3 |^2)*(P`2_3 |^2))) by GROUP_1:def 3
    .= P`3_3*((P`3_3 |^2)*((Q`2_3 |^2)*(Q`3_3)))
    -P`3_3*((Q`3_3*(Q`3_3 |^2))*(P`2_3 |^2)) by GROUP_1:def 3
    .= (P`3_3*(P`3_3 |^2))*((Q`2_3 |^2)*(Q`3_3))
    -P`3_3*(((Q`3_3 |^2)*(Q`3_3))*(P`2_3 |^2)) by GROUP_1:def 3
    .= ((P`3_3 |^2)*(P`3_3))*((Q`2_3 |^2)*(Q`3_3))
    -P`3_3*((P`2_3 |^2)*(Q`3_3 |^(2+1))) by EC_PF_1:24
    .= (P`3_3 |^(2+1))*((Q`2_3 |^2)*(Q`3_3))
    -P`3_3*((P`2_3 |^2)*(Q`3_3 |^3)) by EC_PF_1:24
    .= ((P`3_3*(Q`1_3)) |^3)+(a*((P`3_3*(Q`3_3)) |^2))*(Q`1_3*(P`3_3))
    +b*(P`3_3 |^3)*(Q`3_3 |^3)-(Q`3_3 |^3)*(P`3_3*(P`2_3 |^2))
              by A10,GROUP_1:def 3
    .= (((P`3_3*(Q`1_3)) |^3)+(a*((P`3_3*(Q`3_3)) |^2))*(Q`1_3*(P`3_3)))
    +(b*(P`3_3 |^3)*(Q`3_3 |^3)+((-b*(P`3_3 |^3)*(Q`3_3 |^3))+
    (-(a*((P`3_3*(Q`3_3)) |^2))*(P`1_3*(Q`3_3))-((P`1_3*(Q`3_3)) |^3))))
    by A9,ALGSTR_1:7
    .= (((P`3_3*(Q`1_3)) |^3)+(a*((P`3_3*(Q`3_3)) |^2))*(Q`1_3*(P`3_3)))
    +((b*(P`3_3 |^3)*(Q`3_3 |^3)-b*(P`3_3 |^3)*(Q`3_3 |^3))+
    ((-(a*((P`3_3*(Q`3_3)) |^2))*(P`1_3*(Q`3_3))-((P`1_3*(Q`3_3)) |^3))))
     by ALGSTR_1:7
    .= (((P`3_3*(Q`1_3)) |^3)+(a*((P`3_3*(Q`3_3)) |^2))*(Q`1_3*(P`3_3)))
    +(0.GF(p)+
    (-(a*((P`3_3*(Q`3_3)) |^2))*(P`1_3*(Q`3_3))-((P`1_3*(Q`3_3)) |^3)))
          by VECTSP_1:19
    .= (((P`3_3*(Q`1_3)) |^3)+(a*((P`3_3*(Q`3_3)) |^2))*(Q`1_3*(P`3_3)))
    +(-(a*((P`3_3*(Q`3_3)) |^2))*(P`1_3*(Q`3_3))-((P`1_3*(Q`3_3)) |^3))
    by ALGSTR_1:7
    .= ((P`3_3*(Q`1_3)) |^3)+((a*((P`3_3*(Q`3_3)) |^2))*(Q`1_3*(P`3_3))
    +(-(a*((P`3_3*(Q`3_3)) |^2))*(P`1_3*(Q`3_3))-((P`1_3*(Q`3_3)) |^3)))
    by ALGSTR_1:7
    .= ((P`3_3*(Q`1_3)) |^3)+(((a*((P`3_3*(Q`3_3)) |^2))*(Q`1_3*(P`3_3))
    -(a*((P`3_3*(Q`3_3)) |^2))*(P`1_3*(Q`3_3)))-((P`1_3*(Q`3_3)) |^3))
          by ALGSTR_1:7
    .= ((P`3_3*(Q`1_3)) |^3)+(-((P`1_3*(Q`3_3)) |^3)
    +((a*((P`3_3*(Q`3_3)) |^2))*(Q`1_3*(P`3_3)-P`1_3*(Q`3_3)))) by VECTSP_1:11
    .= (((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3))
    + (a*((P`3_3*(Q`3_3)) |^2))*gf2 by A2,ALGSTR_1:7
    .= (((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3))
    + (a*gf2)*((P`3_3*(Q`3_3)) |^2) by GROUP_1:def 3;
    A12: g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)-gf1*(P`1_3))
    = (((g2*(Q`3_3))*gf1)*(R`3_3))*(gf2*(P`2_3)-gf1*(P`1_3)) by GROUP_1:def 3
    .= ((g2*(Q`3_3))*(gf1*(R`3_3)))*(gf2*(P`2_3)-gf1*(P`1_3)) by GROUP_1:def 3
    .= (gf1*(R`3_3))*((g2*(Q`3_3))*(gf2*(P`2_3)-gf1*(P`1_3))) by GROUP_1:def 3
    .= (gf1*(R`3_3))*(g2*(Q`3_3*(gf2*(P`2_3)-gf1*(P`1_3)))) by GROUP_1:def 3
    .= (gf1*(R`3_3))*(Q`3_3*(gf2*(P`2_3)-gf1*(P`1_3))+
    Q`3_3*(gf2*(P`2_3)-gf1*(P`1_3))) by A1,Th20
    .= (gf1*(R`3_3))*(Q`3_3*(gf2*(P`2_3))-Q`3_3*(gf1*(P`1_3))+
    (gf2*(Q`2_3)-gf1*(Q`1_3))*(P`3_3)) by A5,VECTSP_1:11
    .= (gf1*(R`3_3))*((gf2*(P`2_3))*(Q`3_3)-(gf1*(P`1_3))*(Q`3_3)+
    ((gf2*(Q`2_3))*(P`3_3)-(gf1*(Q`1_3))*(P`3_3))) by VECTSP_1:13
    .= (gf1*(R`3_3))*((gf2*(P`2_3*(Q`3_3))-(gf1*(P`1_3))*(Q`3_3))+
    ((gf2*(Q`2_3))*(P`3_3)-(gf1*(Q`1_3))*(P`3_3))) by GROUP_1:def 3
    .= (gf1*(R`3_3))*(((gf2*(P`2_3*(Q`3_3))-(gf1*(P`1_3))*(Q`3_3))
    +(gf2*(Q`2_3))*(P`3_3))-(gf1*(Q`1_3))*(P`3_3)) by ALGSTR_1:7
    .= (gf1*(R`3_3))*(((gf2*(P`2_3*(Q`3_3))-gf1*(P`1_3*(Q`3_3)))
    +(gf2*(Q`2_3))*(P`3_3))-(gf1*(Q`1_3))*(P`3_3)) by GROUP_1:def 3
    .= (gf1*(R`3_3))*((gf2*(P`2_3*(Q`3_3))
           -gf1*(P`1_3*(Q`3_3))+gf2*(Q`2_3*(P`3_3)))
    -(gf1*(Q`1_3))*(P`3_3)) by GROUP_1:def 3
    .= (gf1*(R`3_3))*((gf2*(P`2_3*(Q`3_3))
            +gf2*(Q`2_3*(P`3_3))-gf1*(P`1_3*(Q`3_3)))
    -(gf1*(Q`1_3))*(P`3_3)) by ALGSTR_1:8
    .= (gf1*(R`3_3))*((gf2*(P`2_3*(Q`3_3))
    +gf2*(Q`2_3*(P`3_3))-gf1*(P`1_3*(Q`3_3)))
    -gf1*(Q`1_3*(P`3_3))) by GROUP_1:def 3
    .= (gf1*(R`3_3))*((gf2*(P`2_3*(Q`3_3))+gf2*(Q`2_3*(P`3_3)))+(
    -gf1*(P`1_3*(Q`3_3))-gf1*(Q`1_3*(P`3_3)))) by ALGSTR_1:7
    .= (gf1*(R`3_3))*(gf2*(P`2_3*(Q`3_3)+Q`2_3*(P`3_3))+(
    (-gf1*(P`1_3*(Q`3_3)))+(-gf1*(Q`1_3*(P`3_3))))) by VECTSP_1:def 7
    .= (gf1*(R`3_3))*(gf2*(P`2_3*(Q`3_3)+Q`2_3*(P`3_3))+(
    (-gf1)*(P`1_3*(Q`3_3))+(-gf1*(Q`1_3*(P`3_3))))) by VECTSP_1:9
    .= (gf1*(R`3_3))*(gf2*(P`2_3*(Q`3_3)+Q`2_3*(P`3_3))+(
    (-gf1)*(P`1_3*(Q`3_3))+(-gf1)*(Q`1_3*(P`3_3)))) by VECTSP_1:9
    .= (gf1*(R`3_3))*(gf2*(P`2_3*(Q`3_3)+Q`2_3*(P`3_3))+
    (-gf1)*(P`1_3*(Q`3_3)+Q`1_3*(P`3_3))) by VECTSP_1:def 7
    .= (gf1*((gf2 |^3)*(P`3_3*(Q`3_3))))*(gf2*(P`2_3*(Q`3_3)+Q`2_3*(P`3_3))+
    (-gf1)*(P`1_3*(Q`3_3)+Q`1_3*(P`3_3))) by GROUP_1:def 3,A3
    .= ((gf1*(gf2 |^3))*(P`3_3*(Q`3_3)))*(gf2*(P`2_3*(Q`3_3)+Q`2_3*(P`3_3))+
    (-gf1)*(P`1_3*(Q`3_3)+Q`1_3*(P`3_3))) by GROUP_1:def 3
    .= (((gf2 |^3)*gf1)*(P`3_3*(Q`3_3)))*(gf2*(P`2_3*(Q`3_3)+Q`2_3*(P`3_3))
    -gf1*(P`1_3*(Q`3_3)+Q`1_3*(P`3_3))) by VECTSP_1:9
    .= ((gf2 |^3)*(gf1*(P`3_3*(Q`3_3))))*(gf2*(P`2_3*(Q`3_3)+Q`2_3*(P`3_3))
    -gf1*(P`1_3*(Q`3_3)+Q`1_3*(P`3_3))) by GROUP_1:def 3
    .= (gf2 |^3)*((gf1*(P`3_3*(Q`3_3)))*(gf2*(P`2_3*(Q`3_3)+Q`2_3*(P`3_3))
    -gf1*(P`1_3*(Q`3_3)+Q`1_3*(P`3_3)))) by GROUP_1:def 3
    .= (gf2 |^3)*((gf1*(P`3_3*(Q`3_3)))*(gf2*(P`2_3*(Q`3_3)+Q`2_3*(P`3_3)))
    -(gf1*(P`3_3*(Q`3_3)))*(gf1*(P`1_3*(Q`3_3)+Q`1_3*(P`3_3)))) by VECTSP_1:11
    .= (gf2 |^3)*(((gf1*(P`3_3*(Q`3_3)))*gf2)*(P`2_3*(Q`3_3)+Q`2_3*(P`3_3))
    -(gf1*(P`3_3*(Q`3_3)))*(gf1*(P`1_3*(Q`3_3)+Q`1_3*(P`3_3))))
          by GROUP_1:def 3
    .= (gf2 |^3)*((gf2*(gf1*(P`3_3*(Q`3_3))))*(P`2_3*(Q`3_3)+Q`2_3*(P`3_3))
    -((gf1*(P`3_3*(Q`3_3)))*gf1)*(P`1_3*(Q`3_3)+Q`1_3*(P`3_3)))
          by GROUP_1:def 3
    .= (gf2 |^3)*(gf2*((gf1*(P`3_3*(Q`3_3)))*(P`2_3*(Q`3_3)+Q`2_3*(P`3_3)))
    -(gf1*(gf1*(P`3_3*(Q`3_3))))*(P`1_3*(Q`3_3)+Q`1_3*(P`3_3)))
            by GROUP_1:def 3
    .= (gf2 |^3)*(gf2*((((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3))
    +(a*gf2)*((P`3_3*(Q`3_3)) |^2))
    -((gf1*gf1)*(P`3_3*(Q`3_3)))*(P`1_3*(Q`3_3)+Q`1_3*(P`3_3)))
     by A11,GROUP_1:def 3
    .= (gf2 |^3)*(gf2*(((((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3)))
    +(a*gf2)*((P`3_3*(Q`3_3)) |^2))
    -((gf1 |^2)*(P`3_3*(Q`3_3)))*(P`1_3*(Q`3_3)+Q`1_3*(P`3_3))) by EC_PF_1:22
    .= (gf2 |^3)*(-((gf1 |^2)*(P`3_3*(Q`3_3)))*(P`1_3*(Q`3_3)+Q`1_3*(P`3_3))
    +(gf2*(((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3))
    +gf2*((a*gf2)*((P`3_3*(Q`3_3)) |^2)))) by VECTSP_1:def 7
    .= (gf2 |^3)*(-((gf1 |^2)*(P`3_3)*(Q`3_3))*(P`1_3*(Q`3_3)+Q`1_3*(P`3_3))
    +(gf2*(((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3))
    +gf2*((a*gf2)*((P`3_3*(Q`3_3)) |^2)))) by GROUP_1:def 3
    .= (gf2 |^3)*(-(P`1_3*(Q`3_3)+Q`1_3*(P`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3))
    +(gf2*(((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3))
    +(gf2*(a*gf2))*((P`3_3*(Q`3_3)) |^2))) by GROUP_1:def 3
    .= (gf2 |^3)*(-(P`1_3*(Q`3_3)+Q`1_3*(P`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3))
    +(gf2*(((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3))
    +(a*(gf2*gf2))*((P`3_3*(Q`3_3)) |^2))) by GROUP_1:def 3
    .= (gf2 |^3)*(-(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3))
    +(gf2*(((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3))
    +(a*(gf2 |^2))*((P`3_3*(Q`3_3)) |^2))) by EC_PF_1:22;
    thus (gf2 |^2)*(P`1_3*(Q`1_3)*(R`3_3)+P`3_3*(Q`1_3)*(R`1_3)
           +P`1_3*(Q`3_3)*(R`1_3))
    + g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3) - gf1*(P`1_3))
    = (gf2 |^3)*(((-gf2)*(((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3))
    +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3)))
    +(-(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3))
    +(gf2*(((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3))
    +(a*(gf2 |^2))*((P`3_3*(Q`3_3)) |^2)))) by A8,A12,VECTSP_1:def 7
    .= (gf2 |^3)*((((-gf2)*(((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3))
    +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3)))
    -(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3)))
    +(gf2*(((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3))
    +(a*(gf2 |^2))*((P`3_3*(Q`3_3)) |^2))) by ALGSTR_1:7
    .= (gf2 |^3)*(((-gf2)*(((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3))
    +((P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3))
    -(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((gf1 |^2)*(P`3_3)*(Q`3_3))))
    +(gf2*(((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3))
    +(a*(gf2 |^2))*((P`3_3*(Q`3_3)) |^2))) by ALGSTR_1:7
    .= (gf2 |^3)*(((-gf2)*(((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3))
    +0.GF(p))
    +(gf2*(((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3))
    +(a*(gf2 |^2))*((P`3_3*(Q`3_3)) |^2))) by VECTSP_1:19
    .= (gf2 |^3)*((-gf2)*(((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3))
    +(gf2*(((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3))
    +(a*(gf2 |^2))*((P`3_3*(Q`3_3)) |^2))) by ALGSTR_1:7
    .= (gf2 |^3)*(((-gf2)*(((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3))
    +gf2*(((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3)))
    +(a*(gf2 |^2))*((P`3_3*(Q`3_3)) |^2)) by ALGSTR_1:7
    .= (gf2 |^3)*((-gf2*(((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3))
    +gf2*(((P`3_3*(Q`1_3)) |^3)-((P`1_3*(Q`3_3)) |^3)))
    +(a*(gf2 |^2))*((P`3_3*(Q`3_3)) |^2)) by VECTSP_1:9
    .= (gf2 |^3)*(0.GF(p)+(a*(gf2 |^2))*((P`3_3*(Q`3_3)) |^2)) by RLVECT_1:5
    .= (gf2 |^3)*((a*(gf2 |^2))*((P`3_3*(Q`3_3)) |^2)) by ALGSTR_1:7
    .= ((gf2 |^3)*(a*(gf2 |^2))*((P`3_3*(Q`3_3)) |^2)) by GROUP_1:def 3
    .= ((a*(gf2 |^2))*(gf2 |^3))*((P`3_3*(Q`3_3))*(P`3_3*(Q`3_3)))
          by EC_PF_1:22
    .= (a*(gf2 |^2))*((gf2 |^3)*((P`3_3*(Q`3_3))*(P`3_3*(Q`3_3))))
          by GROUP_1:def 3
    .= (a*(gf2 |^2))*(((gf2 |^3)*(P`3_3*(Q`3_3)))*(P`3_3*(Q`3_3)))
            by GROUP_1:def 3
    .= (a*(gf2 |^2))*(((gf2 |^3)*(P`3_3)*(Q`3_3))*(P`3_3*(Q`3_3)))
            by GROUP_1:def 3
    .= a*(gf2 |^2)*(P`3_3*(Q`3_3*(R`3_3))) by GROUP_1:def 3,A3
    .= a*(gf2 |^2)*(P`3_3)*(Q`3_3*(R`3_3)) by GROUP_1:def 3
    .= a*(gf2 |^2)*(P`3_3)*(Q`3_3)*(R`3_3) by GROUP_1:def 3;
  end;
