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 Th29:
  1_F+1_F<>0.F & not p _|_ a & not p _|_ x & not q _|_ a & not q
  _|_ x implies ProJ(a,q,p)*ProJ(p,a,x) = ProJ(x,q,p)*ProJ(q,a,x)
proof
  set 0F = 0.F, 1F = 1_F;
  assume that
A1: 1_F+1_F<>0.F and
A2: not p _|_ a and
A3: not p _|_ x and
A4: not q _|_ a and
A5: not q _|_ x;
A6: ( not a _|_ q)& not x _|_ q by A4,A5,Th2;
  ProJ(p,a,x)*ProJ(q,x,a) = ProJ(a,p,q)*ProJ(x,q,p) by A1,A2,A3,A5,Th28;
  then ProJ(p,a,x)*ProJ(q,x,a) = ProJ(a,q,p)"*ProJ(x,q,p) by A2,A4,Th25;
  then
A7: ProJ(a,q,p)*(ProJ(p,a,x)*ProJ(q,x,a)) = (ProJ(a,q,p)*ProJ(a,q,p)")*ProJ
  (x,q,p) by GROUP_1:def 3;
  ProJ(a,q,p) <> 0F by A2,A4,Th23;
  then ProJ(a,q,p)*(ProJ(p,a,x)*ProJ(q,x,a)) = ProJ (x,q,p)*(1F) by A7,
VECTSP_1:def 10;
  then ProJ(a,q,p)*(ProJ(p,a,x)*ProJ(q,x,a)) = ProJ(x,q,p);
  then (ProJ(a,q,p)*ProJ(p,a,x))*ProJ(q,x,a) = ProJ(x,q,p) by GROUP_1:def 3;
  then (ProJ(a,q,p)*ProJ(p,a,x))*ProJ(q,a,x)" = ProJ(x,q,p) by A6,Th25;
  then
A8: (ProJ(a,q,p)*ProJ(p,a,x))*(ProJ(q,a,x)"*ProJ(q,a,x)) = ProJ(x,q,p)*ProJ
  (q,a,x) by GROUP_1:def 3;
  ProJ(q,a,x) <> 0F by A6,Th23;
  then (ProJ(a,q,p)*ProJ(p,a,x))*(1F) = ProJ(x,q,p)*ProJ (q,a,x) by A8,
VECTSP_1:def 10;
  hence thesis;
end;
