reserve FCPS for up-3-dimensional CollProjectiveSpace;
reserve a,a9,b,b9,c,c9,d,d9,o,p,q,r,s,t,u,x,y,z for Element of FCPS;

theorem Th17:
  a<>a9 & o,a,a9 are_collinear & not a,b,c,o are_coplanar & not a9,
  b9,c9 are_collinear & a,b,p are_collinear & a9,b9,p are_collinear & b,c,q
are_collinear & b9,c9,q are_collinear & a,c,r are_collinear &
a9,c9,r are_collinear
  implies p,q,r are_collinear
proof
  assume that
A1: a<>a9 & o,a,a9 are_collinear & not a,b,c,o are_coplanar and
A2: not a9,b9,c9 are_collinear and
A3: a,b,p are_collinear and
A4: a9,b9,p are_collinear and
A5: b,c,q are_collinear & b9,c9,q are_collinear and
A6: a,c,r are_collinear and
A7: a9,c9,r are_collinear;
A8: a,b,c,q are_coplanar & a9,b9,c9,q are_coplanar by A5,Th6;
  c9,r,a9 are_collinear by A7,Th1;
  then
A9: a9,b9,c9,r are_coplanar by Th6;
  p,a9,b9 are_collinear by A4,Th1;
  then
A10: a9,b9,c9,p are_coplanar by Th6;
  c,r,a are_collinear by A6,Th1;
  then
A11: a,b,c,r are_coplanar by Th6;
  p,a,b are_collinear by A3,Th1;
  then
A12: a,b,c,p are_coplanar by Th6;
  ( not a,b,c,a9 are_coplanar)& not a,b,c are_collinear by A1,Th6,Th15;
  hence thesis by A2,A12,A11,A10,A8,A9,Th16;
end;
