reserve PCPP for CollProjectiveSpace,
  a,a9,a1,a2,a3,b,b9,b1,b2,c,c1,c3,d,d9,e,
  o,p,p1,p2,p3,r,q, q1,q2,q3,x,y for Element of PCPP;

theorem
  p<>q & a,b,p are_collinear & a,b,q are_collinear & p,q,r are_collinear
  implies a,b,r are_collinear
proof
  assume that
A1: p<>q and
A2: a,b,p are_collinear & a,b,q are_collinear and
A3: p,q,r are_collinear;
  now
    assume
A4: a<>b;
    b,a,p are_collinear & b,a,q are_collinear by A2,Th1;
    then b,p,q are_collinear by A4,Th2;
    then
A5: p,q,b are_collinear by Th1;
    a,p,q are_collinear by A2,A4,Th2;
    then p,q,a are_collinear by Th1;
    hence thesis by A1,A3,A5,ANPROJ_2:def 8;
  end;
  hence thesis by ANPROJ_2:def 7;
end;
