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 Th9:
  not a1,a2,a3 are_collinear & a1,a2,c3 are_collinear & a2,a3,c1
are_collinear & c1,c3,x are_collinear & c3<>a1 & c3<>a2 & c1<>a2 & c1<>a3
implies
  a1<>x & a3<>x
proof
  assume that
A1: not a1,a2,a3 are_collinear and
A2: a1,a2,c3 are_collinear and
A3: a2,a3,c1 are_collinear and
A4: c1,c3,x are_collinear and
A5: c3<>a1 and
A6: c3<>a2 and
A7: c1<>a2 and
A8: c1<>a3;
A9: a3<>x
  proof
    assume not thesis;
    then
A10: a3,c1,c3 are_collinear by A4,Th1;
    a3,c1,a2 are_collinear by A3,Th1;
    then a3,a2,c3 are_collinear by A8,A10,Th2;
    then
A11: a2,c3,a3 are_collinear by Th1;
    a2,c3,a1 are_collinear by A2,Th1;
    then a2,a1,a3 are_collinear by A6,A11,Th2;
    hence contradiction by A1,Th1;
  end;
  a1<>x
  proof
    assume not thesis;
    then
A12: a1,c3,c1 are_collinear by A4,Th1;
    a1,c3,a2 are_collinear by A2,Th1;
    then a1,c1,a2 are_collinear by A5,A12,Th2;
    then
A13: a2,c1,a1 are_collinear by Th1;
    a2,c1,a3 are_collinear by A3,Th1;
    then a2,a1,a3 are_collinear by A7,A13,Th2;
    hence contradiction by A1,Th1;
  end;
  hence thesis by A9;
end;
