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 Th58:
  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 (gf2 |^2)*(P`3_3 |^2)*(Q`3_3)*
  ((R`2_3 |^2)*(R`3_3) - ((R`1_3 |^3)
    + z`1*(R`1_3)*(R`3_3 |^2) + z`2*(R`3_3 |^3))) = 0.GF(p)
  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 |^2)*(P`3_3 |^2)*(Q`3_3))*((R`2_3 |^2)*(R`3_3))
    = ((gf2 |^2)*(P`3_3 |^2)*(Q`3_3))*(R`2_3 |^2)*(R`3_3) by GROUP_1:def 3
    .= (gf2 |^2)*(P`3_3 |^2)*(R`2_3 |^2)*(Q`3_3)*(R`3_3) by GROUP_1:def 3
    .= ((gf2*(P`3_3)*(R`2_3)) |^2)*(Q`3_3)*(R`3_3) by Th13
    .= ((-(gf1*(R`1_3*(P`3_3)-P`1_3*(R`3_3))
         +gf2*(P`2_3)*(R`3_3))) |^2)*(Q`3_3)*(R`3_3)
    by A1,A2,A3,Th54
    .= ((gf1*(R`1_3*(P`3_3)-P`1_3*(R`3_3))
           +gf2*(P`2_3)*(R`3_3)) |^2)*(Q`3_3)*(R`3_3)
          by Th1
    .= ((gf1*(R`1_3*(P`3_3))-gf1*(P`1_3*(R`3_3))
             +gf2*(P`2_3)*(R`3_3)) |^2)*(Q`3_3)*(R`3_3)
          by VECTSP_1:11
    .= ((gf1*(P`3_3)*(R`1_3)-gf1*(P`1_3*(R`3_3))
               +gf2*(P`2_3)*(R`3_3)) |^2)*(Q`3_3)*(R`3_3)
 by GROUP_1:def 3
    .= ((gf1*(P`3_3)*(R`1_3)-gf1*(P`1_3)*(R`3_3)
     +gf2*(P`2_3)*(R`3_3)) |^2)*(Q`3_3)*(R`3_3)
   by GROUP_1:def 3
    .= ((gf1*(P`3_3)*(R`1_3)+gf2*(P`2_3)*(R`3_3)
            -gf1*(P`1_3)*(R`3_3)) |^2)*(Q`3_3)*(R`3_3)
         by ALGSTR_1:8
    .= ((gf1*(P`3_3)*(R`1_3)+(gf2*(P`2_3)*(R`3_3)
      -gf1*(P`1_3)*(R`3_3))) |^2)*(Q`3_3)*(R`3_3)
   by ALGSTR_1:7
    .= ((gf1*(P`3_3)*(R`1_3)+(gf2*(P`2_3)
          -gf1*(P`1_3))*(R`3_3)) |^2)*(Q`3_3)*(R`3_3)
    by VECTSP_1:13
    .= (((gf1*(P`3_3)) |^2)*(R`1_3 |^2)
          +g2*(gf1*(P`3_3))*(gf2*(P`2_3)-gf1*(P`1_3))
    *(R`1_3)*(R`3_3)
    +((gf2*(P`2_3)-gf1*(P`1_3)) |^2)*(R`3_3 |^2))*(Q`3_3)*(R`3_3)
          by A1,Th27
    .= (((gf1*(P`3_3)) |^2)*(R`1_3 |^2)
           +g2*(gf1*(P`3_3))*(gf2*(P`2_3)-gf1*(P`1_3))
    *(R`1_3)*(R`3_3)+((gf2*(P`2_3)
     -gf1*(P`1_3)) |^2)*(R`3_3 |^2))*(Q`3_3*(R`3_3))
      by GROUP_1:def 3
    .= ((gf1*(P`3_3)) |^2)*(R`1_3 |^2)*(Q`3_3*(R`3_3))
    +g2*(gf1*(P`3_3))*(gf2*(P`2_3)-gf1*(P`1_3))*(R`1_3)*(R`3_3)*(Q`3_3*(R`3_3))
    +((gf2*(P`2_3)-gf1*(P`1_3)) |^2)*(R`3_3 |^2)*(Q`3_3*(R`3_3)) by Th14
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3*(R`3_3))
    +g2*(gf1*(P`3_3))*(gf2*(P`2_3)-gf1*(P`1_3))*(R`1_3)*(R`3_3)*(Q`3_3*(R`3_3))
    +((gf2*(P`2_3)-gf1*(P`1_3)) |^2)*(R`3_3 |^2)*(Q`3_3*(R`3_3)) by BINOM:9
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +g2*(gf1*(P`3_3))*(gf2*(P`2_3)-gf1*(P`1_3))*(R`1_3)*(R`3_3)*(Q`3_3*(R`3_3))
    +((gf2*(P`2_3)-gf1*(P`1_3)) |^2)*(R`3_3 |^2)*(Q`3_3*(R`3_3))
         by GROUP_1:def 3
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +g2*gf1*(P`3_3)*(gf2*(P`2_3)-gf1*(P`1_3))*(R`1_3)*(R`3_3)*(Q`3_3*(R`3_3))
    +((gf2*(P`2_3)-gf1*(P`1_3)) |^2)*(R`3_3 |^2)*(Q`3_3*(R`3_3))
         by GROUP_1:def 3
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +g2*gf1*(P`3_3*(gf2*(P`2_3)-gf1*(P`1_3))*(R`1_3)*(R`3_3)*(Q`3_3*(R`3_3)))
    +((gf2*(P`2_3)-gf1*(P`1_3)) |^2)*(R`3_3 |^2)*(Q`3_3*(R`3_3)) by Th12
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +g2*gf1*((Q`3_3*(R`3_3))*(R`1_3)*(R`3_3)*(P`3_3*(gf2*(P`2_3)-gf1*(P`1_3))))
    +((gf2*(P`2_3)-gf1*(P`1_3)) |^2)*(R`3_3 |^2)*(Q`3_3*(R`3_3)) by Th10
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +g2*gf1*(Q`3_3*(R`3_3))*(R`1_3)*(R`3_3)*(P`3_3*(gf2*(P`2_3)-gf1*(P`1_3)))
    +((gf2*(P`2_3)-gf1*(P`1_3)) |^2)*(R`3_3 |^2)*(Q`3_3*(R`3_3)) by Th11
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +g2*gf1*(Q`3_3)*(R`3_3)*(R`1_3)*(R`3_3)*(P`3_3*(gf2*(P`2_3)-gf1*(P`1_3)))
    +((gf2*(P`2_3)-gf1*(P`1_3)) |^2)*(R`3_3 |^2)*(Q`3_3*(R`3_3))
    by GROUP_1:def 3
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +g2*gf1*(Q`3_3)*(R`3_3)*(R`1_3*(R`3_3)*(P`3_3*(gf2*(P`2_3)-gf1*(P`1_3))))
    +((gf2*(P`2_3)-gf1*(P`1_3)) |^2)*(R`3_3 |^2)*(Q`3_3*(R`3_3)) by Th11
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +g2*gf1*(Q`3_3)*(R`3_3)*((gf2*(P`2_3)-gf1*(P`1_3))*(R`1_3*(R`3_3)*(P`3_3)))
    +((gf2*(P`2_3)-gf1*(P`1_3)) |^2)*(R`3_3 |^2)*(Q`3_3*(R`3_3))
           by GROUP_1:def 3
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)-gf1*(P`1_3))*(R`1_3*(R`3_3)*(P`3_3))
    +((gf2*(P`2_3)-gf1*(P`1_3)) |^2)*(R`3_3 |^2)*(Q`3_3*(R`3_3))
         by GROUP_1:def 3
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)-gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    +((gf2*(P`2_3)-gf1*(P`1_3)) |^2)*(R`3_3 |^2)*(Q`3_3*(R`3_3))
      by GROUP_1:def 3
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)-gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    +((gf2*(P`2_3)-gf1*(P`1_3)) |^2)*(R`3_3 |^2)*(Q`3_3)*(R`3_3)
        by GROUP_1:def 3
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)-gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    +((gf2*(P`2_3)-gf1*(P`1_3)) |^2)*(Q`3_3)*(R`3_3 |^2)*(R`3_3)
        by GROUP_1:def 3
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)-gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    +((gf2*(P`2_3)-gf1*(P`1_3)) |^2)*(Q`3_3)*((R`3_3 |^2)*(R`3_3))
             by GROUP_1:def 3
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)-gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    +((gf2*(P`2_3)-gf1*(P`1_3)) |^2)*(Q`3_3)*(R`3_3 |^(2+1)) by EC_PF_1:24;
    A5: -((gf2 |^2)*(P`3_3 |^2)*(Q`3_3))*(a*(R`1_3)*(R`3_3 |^2))
    = -((gf2 |^2)*(P`3_3*(P`3_3))*(Q`3_3))*(a*(R`1_3)*(R`3_3 |^2))
          by EC_PF_1:22
    .= -((gf2 |^2)*(P`3_3)*(P`3_3)*(Q`3_3))*(a*(R`1_3)*(R`3_3 |^2))
        by GROUP_1:def 3
    .= -((gf2 |^2)*(P`3_3)*(Q`3_3)*(P`3_3))*(a*(R`1_3)*(R`3_3 |^2))
   by GROUP_1:def 3
    .= -((gf2 |^2)*(P`3_3)*(Q`3_3)*(P`3_3))*(a*(R`1_3)*(R`3_3*(R`3_3)))
               by EC_PF_1:22
    .= -((gf2 |^2)*(P`3_3)*(Q`3_3)*(P`3_3))*(a*(R`1_3)*(R`3_3)*(R`3_3))
        by GROUP_1:def 3
    .= -((gf2 |^2)*(P`3_3)*(Q`3_3)*(P`3_3))*(a*(R`3_3)*(R`1_3)*(R`3_3))
            by GROUP_1:def 3
    .= -((gf2 |^2)*(P`3_3)*(Q`3_3))*(P`3_3*(a*(R`3_3)*(R`1_3)*(R`3_3)))
             by GROUP_1:def 3
    .= -((gf2 |^2)*(P`3_3)*(Q`3_3))*(P`3_3*(a*(R`3_3))*(R`1_3)*(R`3_3))
          by Th11
    .= -((gf2 |^2)*(P`3_3)*(Q`3_3))*((a*(R`3_3))*(P`3_3*(R`1_3)*(R`3_3)))
         by Th11
    .= -((gf2 |^2)*(P`3_3)*(Q`3_3)*(a*(R`3_3)))*(P`3_3*(R`1_3)*(R`3_3))
          by GROUP_1:def 3
    .= -((gf2 |^2)*(P`3_3)*(Q`3_3)*a*(R`3_3))*(R`1_3*(P`3_3)*(R`3_3))
             by GROUP_1:def 3
    .= -((gf2 |^2)*(P`3_3*(Q`3_3)*a*(R`3_3)))*(P`3_3*(R`1_3)*(R`3_3))
           by Th11
    .= -((gf2 |^2)*(a*(P`3_3)*(Q`3_3)*(R`3_3)))*(P`3_3*(R`1_3)*(R`3_3))
           by GROUP_1:def 3
    .= -((gf2 |^2)*a*(P`3_3)*(Q`3_3)*(R`3_3))*(P`3_3*(R`1_3)*(R`3_3))
          by Th11
    .= -((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)))*(P`3_3*(R`1_3)*(R`3_3))
      by A1,A2,A3,Th57
    .= -((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))
    *(P`3_3*(R`1_3)*(R`3_3))+g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
                -gf1*(P`1_3))*(P`3_3*(R`1_3)*(R`3_3)))
    by VECTSP_1:def 7
    .= -((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)))
    *(P`3_3*(R`1_3)*(R`3_3))+g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
      -gf1*(P`1_3))*(P`3_3*(R`1_3)*(R`3_3)))
    by ALGSTR_1:7
    .= -((gf2 |^2)*(P`3_3*(R`1_3)*(R`3_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)))+g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
     -gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3)))
    by GROUP_1:def 3
    .= -g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
    -gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    -(gf2 |^2)*(P`3_3*(R`1_3)*(R`3_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 VECTSP_1:17
    .= -g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
       -gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    -((gf2 |^2)*(P`3_3*(R`1_3)*(R`3_3))*(P`1_3*(Q`1_3)*(R`3_3))
                +(gf2 |^2)*(P`3_3*(R`1_3)*(R`3_3))*
    (P`3_3*(Q`1_3)*(R`1_3)+P`1_3*(Q`3_3)*(R`1_3))) by VECTSP_1:def 7
    .= -g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
      -gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    -((gf2 |^2)*(P`3_3*(R`1_3)*(R`3_3))*(P`1_3*(Q`1_3)*(R`3_3))
     +(gf2 |^2)*(P`3_3*(R`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
    .= -g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
     -gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    -((gf2 |^2)*(P`3_3*(R`1_3)*(R`3_3))*(P`1_3*(Q`1_3)*(R`3_3))
     +(gf2 |^2)*(P`3_3*(R`1_3)*(R`3_3))*
    (P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`1_3)) by GROUP_1:def 3
    .= -g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
          -gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    -((gf2 |^2)*(P`3_3*(R`1_3)*(R`3_3))*(P`1_3*(Q`1_3)*(R`3_3))
    +(gf2 |^2)*(P`3_3)*(R`1_3)*(R`3_3)*
    (P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`1_3)) by Th12
    .= -g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
           -gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    -((gf2 |^2)*(P`3_3*(R`1_3)*(R`3_3))*(P`1_3*(Q`1_3)*(R`3_3))
    +(gf2 |^2)*(P`3_3)*(R`1_3*(R`3_3)*
    (P`3_3*(Q`1_3)+P`1_3*(Q`3_3)))*(R`1_3)) by Th11
    .= -g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
    -gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    -((gf2 |^2)*(P`3_3*(R`1_3)*(R`3_3))*(P`1_3*(Q`1_3)*(R`3_3))
          +(gf2 |^2)*(P`3_3)*(
    (P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`1_3)*(R`3_3))*(R`1_3)) by GROUP_1:def 3
    .= -g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
       -gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    -((gf2 |^2)*(P`3_3*(R`1_3)*(R`3_3))*(P`1_3*(Q`1_3)*(R`3_3))
        +(gf2 |^2)*(P`3_3)*
    (P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`1_3)*(R`3_3)*(R`1_3)) by Th11
    .= -g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
       -gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    -((gf2 |^2)*(P`3_3*(R`1_3)*(R`3_3))*
     (P`1_3*(Q`1_3)*(R`3_3))+(gf2 |^2)*(P`3_3)*
    (P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`1_3*(R`3_3)*(R`1_3))) by Th11
    .= -g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
       -gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    -((gf2 |^2)*(P`3_3*(R`1_3)*(R`3_3))*(P`1_3*(Q`1_3)*(R`3_3))
    +(gf2 |^2)*(P`3_3)*
    (P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`1_3*(R`1_3)*(R`3_3))) by GROUP_1:def 3
    .= -g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
       -gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    -((gf2 |^2)*(P`3_3*(R`1_3)*(R`3_3))*(P`1_3*(Q`1_3)*(R`3_3))
           +(gf2 |^2)*(P`3_3)*
    (P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((R`1_3 |^2)*(R`3_3))) by EC_PF_1:22
    .= -g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
      -gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    -((gf2 |^2)*(P`3_3)*(R`1_3)*(R`3_3)*(P`1_3*(Q`1_3)*(R`3_3))
      +(gf2 |^2)*(P`3_3)*
    (P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((R`1_3 |^2)*(R`3_3))) by Th11
    .= -g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
         -gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    -((gf2 |^2)*(P`3_3)*(R`1_3*(R`3_3)*(P`1_3*(Q`1_3)*(R`3_3)))
          +(gf2 |^2)*(P`3_3)*
    (P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((R`1_3 |^2)*(R`3_3))) by Th11
    .= -g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
           -gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    -((gf2 |^2)*(P`3_3)*((P`1_3*(Q`1_3))*(R`3_3*(R`1_3*(R`3_3))))
    +(gf2 |^2)*(P`3_3)*
    (P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((R`1_3 |^2)*(R`3_3))) by GROUP_1:def 3
    .= -g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
         -gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    -((gf2 |^2)*(P`3_3)*(P`1_3*(Q`1_3)*(R`1_3*(R`3_3*(R`3_3))))
       +(gf2 |^2)*(P`3_3)*
    (P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((R`1_3 |^2)*(R`3_3))) by GROUP_1:def 3
    .= -g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
     -gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    -((gf2 |^2)*(P`3_3)*(P`1_3*(Q`1_3)*(R`1_3*(R`3_3 |^2)))+(gf2 |^2)*(P`3_3)*
    (P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((R`1_3 |^2)*(R`3_3))) by EC_PF_1:22
    .= -g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
      -gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    -((gf2 |^2)*(P`3_3)*(P`1_3*(Q`1_3)*(R`1_3)*(R`3_3 |^2))+(gf2 |^2)*(P`3_3)*
    (P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((R`1_3 |^2)*(R`3_3))) by GROUP_1:def 3
    .= -g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
       -gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    -((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3)*(R`3_3 |^2)+(gf2 |^2)*(P`3_3)*
    (P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((R`1_3 |^2)*(R`3_3))) by Th11
    .= -g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
       -gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    -((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2)
    -(gf2 |^2)*(P`3_3)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*((R`1_3 |^2)*(R`3_3))
                by VECTSP_1:17
    .= -g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
       -gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    -((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2)
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(P`3_3)*((R`1_3 |^2)*(R`3_3))
            by GROUP_1:def 3
    .= -g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
      -gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    -((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2)
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(P`3_3*((R`1_3 |^2)*(R`3_3)))
                 by GROUP_1:def 3
    .= -g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
      -gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    -((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2)
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3*((R`1_3 |^2)*(P`3_3)))
          by GROUP_1:def 3
    .= -g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
      -gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    -((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2)
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)*((R`1_3 |^2)*(P`3_3))
          by GROUP_1:def 3;
    A6: -((gf2 |^2)*(P`3_3 |^2)*(Q`3_3))*(b*(R`3_3 |^3))
    = -(gf2 |^2)*(P`3_3 |^2)*(Q`3_3)*b*(R`3_3 |^(2+1)) by GROUP_1:def 3
    .= -(gf2 |^2)*(P`3_3 |^2)*(Q`3_3)*b*((R`3_3 |^2)*(R`3_3)) by EC_PF_1:24
    .= -b*((gf2 |^2)*(P`3_3 |^2)*(Q`3_3))*(R`3_3)*(R`3_3 |^2) by GROUP_1:def 3
    .= -b*(gf2 |^2)*(P`3_3 |^2)*(Q`3_3)*(R`3_3)*(R`3_3 |^2) by Th12
    .= -(-(gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3)
            +((gf2*(P`2_3) - gf1*(P`1_3)) |^2)*(Q`3_3)*(R`3_3))
    *(R`3_3 |^2) by A1,A2,A3,Th56
    .= (-(-(gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3)
     +((gf2*(P`2_3) - gf1*(P`1_3)) |^2)*(Q`3_3)*(R`3_3)))
    *(R`3_3 |^2) by VECTSP_1:9
    .= (-((gf2*(P`2_3) - gf1*(P`1_3)) |^2)*(Q`3_3)*(R`3_3)
      +(gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))
    *(R`3_3 |^2) by VECTSP_1:33
    .= (-((gf2*(P`2_3) - gf1*(P`1_3)) |^2)*(Q`3_3)*(R`3_3))*(R`3_3 |^2)
    +((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2) by VECTSP_1:def 7

    .= -((gf2*(P`2_3) - gf1*(P`1_3)) |^2)*(Q`3_3)*(R`3_3)*(R`3_3 |^2)
    +((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2) by VECTSP_1:9
    .= -((gf2*(P`2_3) - gf1*(P`1_3)) |^2)*(Q`3_3)*(R`3_3*(R`3_3 |^2))
    +((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2) by GROUP_1:def 3
    .=-((gf2*(P`2_3) - gf1*(P`1_3)) |^2)*(Q`3_3)*(R`3_3 |^(2+1))
    +((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2) by EC_PF_1:24;
    A7: ((gf2 |^2)*(P`3_3 |^2)*(Q`3_3))*((R`2_3 |^2)*(R`3_3))
    -((gf2 |^2)*(P`3_3 |^2)*(Q`3_3))*(b*(R`3_3 |^3))
    = (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)-gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    +(((gf2*(P`2_3)-gf1*(P`1_3)) |^2)*(Q`3_3)*(R`3_3 |^3)
    +(-((gf2*(P`2_3) - gf1*(P`1_3)) |^2)*(Q`3_3)*(R`3_3 |^3)
    +((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2)))
            by A4,A6,ALGSTR_1:7
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)-gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    +(((gf2*(P`2_3)-gf1*(P`1_3)) |^2)*(Q`3_3)*(R`3_3 |^3)
    -((gf2*(P`2_3) - gf1*(P`1_3)) |^2)*(Q`3_3)*(R`3_3 |^3)
    +((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2)) by ALGSTR_1:7
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)
        -gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))+(0.GF(p)
    +((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2)) by VECTSP_1:19
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)-gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    +((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2) by RLVECT_1:4
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2)
    +g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)-gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
     by ALGSTR_1:8;
    A8: ((gf2 |^2)*(P`3_3 |^2)*(Q`3_3))*((R`2_3 |^2)*(R`3_3))
    -((gf2 |^2)*(P`3_3 |^2)*(Q`3_3))*(b*(R`3_3 |^3))
    -((gf2 |^2)*(P`3_3 |^2)*(Q`3_3))*(a*(R`1_3)*(R`3_3 |^2))
    = (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2)
    +(g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)-gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    +(-g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)-gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    -((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2)
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)*((R`1_3 |^2)*(P`3_3))))
         by A5,A7,ALGSTR_1:7
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2)
    +(g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)-gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    +(-g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)-gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    +(-((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2)
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)*((R`1_3 |^2)*(P`3_3)))))
    by ALGSTR_1:7
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2)
    +((g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)-gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3))
    -g2*gf1*(Q`3_3)*(R`3_3)*(gf2*(P`2_3)-gf1*(P`1_3))*(R`1_3*(P`3_3)*(R`3_3)))
    +(-((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2)
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)*((R`1_3 |^2)*(P`3_3))))
          by ALGSTR_1:7
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2)
    +(0.GF(p)+(-((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2)
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)*((R`1_3 |^2)*(P`3_3))))
    by VECTSP_1:19
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2)
    +(-((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2)
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)*((R`1_3 |^2)*(P`3_3)))
         by RLVECT_1:4
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +(((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2)
    +(-((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2)
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)*((R`1_3 |^2)*(P`3_3))))
           by ALGSTR_1:7
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +(((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2)
    -((gf2 |^2)*(P`3_3)*(P`1_3)*(Q`1_3)*(R`1_3))*(R`3_3 |^2)
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)*((R`1_3 |^2)*(P`3_3)))
            by ALGSTR_1:7
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    +(0.GF(p)-(gf2 |^2)*(P`3_3*(Q`1_3)
    +P`1_3*(Q`3_3))*(R`3_3)*((R`1_3 |^2)*(P`3_3)))
   by VECTSP_1:19
    .= (gf1 |^2)*(P`3_3 |^2)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)*((R`1_3 |^2)*(P`3_3))
    by RLVECT_1:14
    .= (gf1 |^2)*(P`3_3*(P`3_3))*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)*((R`1_3 |^2)*(P`3_3))
   by EC_PF_1:22
    .= (gf1 |^2)*(P`3_3)*(P`3_3)*(R`1_3 |^2)*(Q`3_3)*(R`3_3)
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)*((R`1_3 |^2)*(P`3_3))
  by GROUP_1:def 3
    .= (gf1 |^2)*(P`3_3)*(P`3_3*(R`1_3 |^2)*(Q`3_3)*(R`3_3))
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)*((R`1_3 |^2)*(P`3_3))
      by Th11
    .= (gf1 |^2)*(P`3_3)*(P`3_3*(R`1_3 |^2)*(Q`3_3*(R`3_3)))
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)*((R`1_3 |^2)*(P`3_3))
          by GROUP_1:def 3
    .= (gf1 |^2)*(P`3_3)*(Q`3_3*(R`3_3))*((R`1_3 |^2)*(P`3_3))
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)*((R`1_3 |^2)*(P`3_3))
   by GROUP_1:def 3
    .= (gf1 |^2)*(P`3_3)*(Q`3_3)*(R`3_3)*((R`1_3 |^2)*(P`3_3))
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)*((R`1_3 |^2)*(P`3_3))
   by GROUP_1:def 3;
    thus (gf2 |^2)*(P`3_3 |^2)*(Q`3_3)*
    ((R`2_3 |^2)*(R`3_3) -
    ((R`1_3 |^3) + z`1*(R`1_3)*(R`3_3 |^2) + z`2*(R`3_3 |^3)))
    = (gf2 |^2)*(P`3_3 |^2)*(Q`3_3)*((R`2_3 |^2)*(R`3_3))
    -(gf2 |^2)*(P`3_3 |^2)*(Q`3_3)*((R`1_3 |^3)
    + a*(R`1_3)*(R`3_3 |^2) + b*(R`3_3 |^3))
    by VECTSP_1:11
    .= (gf2 |^2)*(P`3_3 |^2)*(Q`3_3)*((R`2_3 |^2)*(R`3_3))
    -((gf2 |^2)*(P`3_3 |^2)*(Q`3_3)*(R`1_3 |^3)
    +(gf2 |^2)*(P`3_3 |^2)*(Q`3_3)*(a*(R`1_3)*(R`3_3 |^2))
    +(gf2 |^2)*(P`3_3 |^2)*(Q`3_3)*(b*(R`3_3 |^3))) by Th14
    .= (gf2 |^2)*(P`3_3 |^2)*(Q`3_3)*((R`2_3 |^2)*(R`3_3))
    -((gf2 |^2)*(P`3_3 |^2)*(Q`3_3)*(R`1_3 |^3)
    +((gf2 |^2)*(P`3_3 |^2)*(Q`3_3)*(a*(R`1_3)*(R`3_3 |^2))
    +(gf2 |^2)*(P`3_3 |^2)*(Q`3_3)*(b*(R`3_3 |^3)))) by ALGSTR_1:7
    .= ((gf2 |^2)*(P`3_3 |^2)*(Q`3_3)*((R`2_3 |^2)*(R`3_3))
    -((gf2 |^2)*(P`3_3 |^2)*(Q`3_3)*(a*(R`1_3)*(R`3_3 |^2))
    +(gf2 |^2)*(P`3_3 |^2)*(Q`3_3)*(b*(R`3_3 |^3)))
    -(gf2 |^2)*(P`3_3 |^2)*(Q`3_3)*(R`1_3 |^3)) by VECTSP_1:17
    .= ((gf2 |^2)*(P`3_3 |^2)*(Q`3_3)*((R`2_3 |^2)*(R`3_3))
    -(gf2 |^2)*(P`3_3 |^2)*(Q`3_3)*(b*(R`3_3 |^3))
    -(gf2 |^2)*(P`3_3 |^2)*(Q`3_3)*(a*(R`1_3)*(R`3_3 |^2)))
    -(gf2 |^2)*(P`3_3 |^2)*(Q`3_3)*(R`1_3 |^3) by VECTSP_1:17
    .= (gf1 |^2)*(P`3_3)*(Q`3_3)*(R`3_3)*((R`1_3 |^2)*(P`3_3))
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)*((R`1_3 |^2)*(P`3_3))
    -(gf2 |^2)*(P`3_3*(P`3_3))*(Q`3_3)*(R`1_3 |^3) by A8,EC_PF_1:22
    .= (gf1 |^2)*(P`3_3)*(Q`3_3)*(R`3_3)*((R`1_3 |^2)*(P`3_3))
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)*((R`1_3 |^2)*(P`3_3))
    -(gf2 |^2)*(P`3_3)*(P`3_3)*(Q`3_3)*(R`1_3 |^3) by GROUP_1:def 3
    .= (gf1 |^2)*(P`3_3)*(Q`3_3)*(R`3_3)*((R`1_3 |^2)*(P`3_3))
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)*((R`1_3 |^2)*(P`3_3))
    -(gf2 |^2)*(P`3_3)*(Q`3_3)*(P`3_3)*(R`1_3 |^3) by GROUP_1:def 3
    .= (gf1 |^2)*(P`3_3)*(Q`3_3)*(R`3_3)*((R`1_3 |^2)*(P`3_3))
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)*((R`1_3 |^2)*(P`3_3))
    -(gf2 |^2)*(P`3_3)*(Q`3_3)*(R`1_3 |^(2+1))*(P`3_3) by GROUP_1:def 3
    .= (gf1 |^2)*(P`3_3)*(Q`3_3)*(R`3_3)*((R`1_3 |^2)*(P`3_3))
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)*((R`1_3 |^2)*(P`3_3))
    -(gf2 |^2)*(P`3_3)*(Q`3_3)*((R`1_3 |^2)*(R`1_3))*(P`3_3) by EC_PF_1:24
    .= (gf1 |^2)*(P`3_3)*(Q`3_3)*(R`3_3)*((R`1_3 |^2)*(P`3_3))
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)*((R`1_3 |^2)*(P`3_3))
    -(gf2 |^2)*(P`3_3)*(Q`3_3)*(((R`1_3 |^2)*(R`1_3))*(P`3_3)) by GROUP_1:def 3
    .= (gf1 |^2)*(P`3_3)*(Q`3_3)*(R`3_3)*((R`1_3 |^2)*(P`3_3))
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)*((R`1_3 |^2)*(P`3_3))
    -(gf2 |^2)*(P`3_3)*(Q`3_3)*(R`1_3*((R`1_3 |^2)*(P`3_3))) by GROUP_1:def 3
    .= (gf1 |^2)*(P`3_3)*(Q`3_3)*(R`3_3)*((R`1_3 |^2)*(P`3_3))
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)*((R`1_3 |^2)*(P`3_3))
    -(gf2 |^2)*(P`3_3)*(Q`3_3)*(R`1_3)*((R`1_3 |^2)*(P`3_3)) by GROUP_1:def 3
    .= ((gf1 |^2)*(P`3_3)*(Q`3_3)*(R`3_3)
    -(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)
    -(gf2 |^2)*(P`3_3)*(Q`3_3)*(R`1_3))*((R`1_3 |^2)*(P`3_3)) by Th14
    .= ((gf1 |^2)*(P`3_3)*(Q`3_3)*(R`3_3)-((gf2 |^2)*(P`3_3)*(Q`3_3)*(R`1_3)
    +(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)))*((R`1_3 |^2)*(P`3_3))
    by VECTSP_1:17
    .= ((gf1 |^2)*(P`3_3)*(Q`3_3)*(R`3_3)-((gf2 |^2)*(P`3_3*(Q`3_3)*(R`1_3))
    +(gf2 |^2)*(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)))*((R`1_3 |^2)*(P`3_3))
             by Th11
    .= ((gf1 |^2)*(P`3_3)*(Q`3_3)*(R`3_3)-((gf2 |^2)*(P`3_3*(Q`3_3)*(R`1_3))
    +(gf2 |^2)*((P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3))))*((R`1_3 |^2)*(P`3_3))
   by GROUP_1:def 3
    .= ((gf1 |^2)*(P`3_3)*(Q`3_3)*(R`3_3)-(gf2 |^2)*(P`3_3*(Q`3_3)*(R`1_3)
    +(P`3_3*(Q`1_3)+P`1_3*(Q`3_3))*(R`3_3)))*((R`1_3 |^2)*(P`3_3))
       by VECTSP_1:def 7
    .= (-(gf2 |^2)*(P`3_3*(Q`3_3)*(R`1_3)
     +(P`3_3*(Q`1_3)*(R`3_3)+P`1_3*(Q`3_3)*(R`3_3)))
    +(gf1 |^2)*(P`3_3)*(Q`3_3)*(R`3_3))*((R`1_3 |^2)*(P`3_3)) by VECTSP_1:def 7

    .= (-(gf2 |^2)*(P`3_3*(Q`3_3)*(R`1_3)
         +P`3_3*(Q`1_3)*(R`3_3)+P`1_3*(Q`3_3)*(R`3_3))
    +(gf1 |^2)*(P`3_3)*(Q`3_3)*(R`3_3))*((R`1_3 |^2)*(P`3_3)) by ALGSTR_1:7
    .= (-(gf2 |^2)*(P`3_3*(Q`3_3)*(R`1_3)
       +P`3_3*(Q`1_3)*(R`3_3)+P`1_3*(Q`3_3)*(R`3_3))
    +(gf1 |^2)*(P`3_3*(Q`3_3)*(R`3_3)))*((R`1_3 |^2)*(P`3_3)) by Th11
    .= 0.GF(p)*((R`1_3 |^2)*(P`3_3)) by A1,A2,A3,Th55
    .= 0.GF(p);
  end;
