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 Th24:
  not b _|_ a & not q _|_ a implies ProJ(a,b,p)*ProJ(a,b,q)" = ProJ(a,q,p)
proof
  assume that
A1: not b _|_ a and
A2: not q _|_ a;
  ProJ(a,q,p)*q- ProJ(a,b,ProJ(a,q,p)*q)*b _|_ a & p-ProJ(a,q,p)*q _|_ a
  by A1,A2,Th14;
  then (p+(-ProJ(a,q,p)*q))+(ProJ(a,q,p)*q-(ProJ(a,b,ProJ(a,q,p)*q)*b)) _|_ a
  by Def1;
  then ((p+(-ProJ(a,q,p)*q))+(ProJ(a,q,p)*q))+(-(ProJ(a,b,ProJ(a,q,p)*q))*b)
  _|_ a by RLVECT_1:def 3;
  then (p+((-ProJ(a,q,p)*q)+(ProJ(a,q,p)*q))+(-(ProJ(a,b,ProJ(a,q,p)*q))*b))
  _|_ a by RLVECT_1:def 3;
  then p+0.S+(-(ProJ(a,b,ProJ(a,q,p)*q))*b) _|_ a by RLVECT_1:5;
  then p+(-(ProJ(a,b,ProJ(a,q,p)*q))*b) _|_ a by RLVECT_1:4;
  then
A3: p-(ProJ(a,q,p)*ProJ(a,b,q))*b _|_ a by A1,Th15;
  p-ProJ(a,b,p)*b _|_ a by A1,Th14;
  then ProJ(a,q,p)*ProJ(a,b,q) = ProJ(a,b,p) by A1,A3,Th12;
  then
A4: ProJ(a,q,p)*(ProJ(a,b,q)*ProJ(a,b,q)") = ProJ(a,b,p)*ProJ (a,b,q)" by
GROUP_1:def 3;
  ProJ(a,b,q) <> 0.F by A1,A2,Th23;
  then ProJ(a,q,p)*1_F = ProJ(a,b,p)*ProJ(a,b,q)" by A4,VECTSP_1:def 10;
  hence thesis;
end;
