reserve F for Field;
reserve S for SymSp of F;
reserve a,b,c,d,a9,b9,p,q,r,s,x,y,z for Element of S;
reserve k,l for Element of F;

theorem Th28:
  1_F+1_F<>0.F & not a _|_ p & not a _|_ q & not b _|_ q implies
  ProJ(a,p,q)*ProJ(b,q,p) = ProJ(p,a,b)*ProJ(q,b,a)
proof
  assume that
A1: 1_F+1_F<>0.F and
A2: not a _|_ p and
A3: not a _|_ q and
A4: not b _|_ q;
A5: now
    assume that
A6: p _|_ q and
A7: a _|_ b;
    not q _|_ b by A4,Th2;
    then
A8: ProJ(b,q,p) = ProJ (b,q,p+a) by A7,Th19;
A9: not p+a _|_ q by A3,A6,Th4;
    then
A10: ProJ(b,q,p+a) = (-ProJ(q,p+a,b)")*ProJ(p+a,q,b) by A4,Th27;
A11: a _|_ a by A1,Th13;
A12: not p _|_ a by A2,Th2;
    then
A13: ProJ(a,p,q) = ProJ(a,p+a,q) by A11,Th19;
    not p+a _|_ a by A12,A11,Th4;
    then
A14: not a _|_ p+a by Th2;
A15: not q _|_ p+a by A9,Th2;
    then ProJ(a,p+a,q) = (-ProJ(p+a,q,a)")*ProJ(q,p+a,a) by A14,Th27;
    then
    ProJ(a,p,q)*ProJ(b,q,p) = ((ProJ(q,p+a,a)*(-ProJ(p+a,q,a)"))*(-ProJ(q
    ,p+a,b)"))*ProJ (p+a,q,b) by A13,A8,A10,GROUP_1:def 3
      .= (ProJ(q,p+a,a)*((-ProJ(p+a,q,a)")*(-ProJ(q,p+a,b)")))*ProJ (p+a,q,b
    ) by GROUP_1:def 3
      .= (ProJ(q,p+a,a)*(ProJ(q,p+a,b)"*ProJ(p+a,q,a)"))*ProJ(p+a,q,b) by
VECTSP_1:10
      .= ((ProJ(q,p+a,a)*ProJ(q,p+a,b)")*ProJ(p+a,q,a)")*ProJ (p+a,q,b) by
GROUP_1:def 3
      .= (ProJ(q,b,a)*ProJ(p+a,q,a)")*ProJ(p+a,q,b) by A4,A9,Th24
      .= ProJ(q,b,a)*(ProJ(p+a,q,b)*ProJ(p+a,q,a)") by GROUP_1:def 3
      .= ProJ(q,b,a)*ProJ(p+a,a,b) by A14,A15,Th24
      .= ProJ(q,b,a)*ProJ(p,a,b) by A2,A7,A11,Th20;
    hence thesis;
  end;
A16: now
    assume
A17: not a _|_ b;
A18: not q _|_ b by A4,Th2;
    then
A19: ProJ(q,b,a) = (-ProJ(b,a,q)")*ProJ(a,b,q) by A17,Th27;
A20: not p _|_ a by A2,Th2;
A21: not b _|_ a by A17,Th2;
    then ProJ(p,a,b) = (-ProJ(a,b,p)")*ProJ(b,a,p) by A20,Th27;
    then
    ProJ(p,a,b)*ProJ(q,b,a) = ((ProJ(b,a,p)*(-ProJ(a,b,p)"))*(-ProJ(b,a,q
    )"))*ProJ (a,b,q) by A19,GROUP_1:def 3
      .= (ProJ(b,a,p)*((-ProJ(a,b,p)")*(-ProJ(b,a,q)")))*ProJ (a,b,q) by
GROUP_1:def 3
      .= (ProJ(b,a,p)*(ProJ(b,a,q)"*ProJ(a,b,p)"))*ProJ(a,b,q) by VECTSP_1:10
      .= ((ProJ(b,a,p)*ProJ(b,a,q)")*ProJ(a,b,p)")*ProJ(a,b,q) by GROUP_1:def 3
      .= (ProJ(b,q,p)*ProJ(a,b,p)")*ProJ(a,b,q) by A17,A18,Th24
      .= ProJ(b,q,p)*(ProJ(a,b,q)*ProJ(a,b,p)") by GROUP_1:def 3
      .= ProJ(b,q,p)*ProJ(a,p,q) by A21,A20,Th24;
    hence thesis;
  end;
  now
    assume
A22: not p _|_ q;
    then
A23: ProJ(b,q,p) = (-(ProJ(q,p,b)"))*ProJ(p,q,b) by A4,Th27;
A24: not q _|_ p by A22,Th2;
    then ProJ(a,p,q) = (-ProJ(p,q,a)")*ProJ(q,p,a) by A2,Th27;
    then
    ProJ(a,p,q)*ProJ(b,q,p) = ((ProJ(q,p,a)*(-ProJ(p,q,a)"))*(-ProJ(q,p,b
    )"))*ProJ(p,q,b) by A23,GROUP_1:def 3
      .= (ProJ(q,p,a)*((-ProJ(p,q,a)")*(-ProJ(q,p,b)")))*ProJ (p,q,b) by
GROUP_1:def 3
      .= (ProJ(q,p,a)*(ProJ(q,p,b)"*ProJ(p,q,a)"))*ProJ(p,q,b) by VECTSP_1:10
      .= ((ProJ(q,p,a)*ProJ(q,p,b)")*ProJ(p,q,a)")*ProJ(p,q,b) by GROUP_1:def 3
      .= (ProJ(q,b,a)*ProJ(p,q,a)")*ProJ(p,q,b) by A4,A22,Th24
      .= ProJ(q,b,a)*(ProJ(p,q,b)*ProJ(p,q,a)") by GROUP_1:def 3
      .= ProJ(q,b,a)*ProJ(p,a,b) by A2,A24,Th24;
    hence thesis;
  end;
  hence thesis by A16,A5;
end;
