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 Th5:
  not o,a,d are_collinear & o,d,d9 are_collinear & d<>d9 & a9,d9,s
  are_collinear & o,a,a9 are_collinear & o<>a9 implies s<>d
proof
  assume that
A1: not o,a,d are_collinear and
A2: o,d,d9 are_collinear and
A3: d<>d9 and
A4: a9,d9,s are_collinear and
A5: o,a,a9 are_collinear and
A6: o<>a9;
  assume not thesis;
  then
A7: d,d9,a9 are_collinear by A4,Th1;
  d,d9,o are_collinear by A2,Th1;
  then d,o,a9 are_collinear by A3,A7,COLLSP:6;
  then
A8: o,a9,d are_collinear by Th1;
  o,a9,a are_collinear by A5,Th1;
  hence contradiction by A1,A6,A8,COLLSP:6;
end;
