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 Th19:
  not a,b,c,o are_coplanar & o,a,a9 are_collinear & o,b,b9
are_collinear & o,c,c9 are_collinear & o<>a9 & o<>b9 & o<>c9
implies not a9,b9,c9
  are_collinear & not a9,b9,c9,o are_coplanar
proof
  assume that
A1: ( not a,b,c,o are_coplanar)& o,a,a9 are_collinear and
A2: o,b,b9 are_collinear and
A3: o,c,c9 are_collinear and
A4: o<>a9 and
A5: o<>b9 and
A6: o<>c9;
  a,o,a9 are_collinear & not o,b,c,a are_coplanar by A1,Th1,Th7;
  then not o,b,c,a9 are_coplanar by A4,Th15;
  then
A7: not o,a9,b,c are_coplanar by Th7;
  c,o,c9 are_collinear by A3,Th1;
  then not o,a9,b,c9 are_coplanar by A6,A7,Th15;
  then
A8: not o,a9,c9,b are_coplanar by Th7;
  b,o,b9 are_collinear by A2,Th1;
  then not o,a9,c9,b9 are_coplanar by A5,A8,Th15;
  then not a9,b9,c9,o are_coplanar by Th7;
  hence thesis by Th6;
end;
