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 Th20:
  a,b,c,o are_coplanar & not a,b,c,d are_coplanar & not a,b,d,o
are_coplanar & o,d,d9 are_collinear & o,a,a9 are_collinear &
o,b,b9 are_collinear
  & a,d,s are_collinear & a9,d9,s are_collinear & b,d,t are_collinear & b9,d9,t
  are_collinear & c,d,u are_collinear & o<>a9 & d<>d9 & o<>b9 implies not s,t,u
  are_collinear
proof
  assume that
A1: a,b,c,o are_coplanar and
A2: not a,b,c,d are_coplanar and
A3: not a,b,d,o are_coplanar and
A4: o,d,d9 are_collinear and
A5: o,a,a9 are_collinear and
A6: o,b,b9 are_collinear and
A7: a,d,s are_collinear and
A8: a9,d9,s are_collinear and
A9: b,d,t are_collinear and
A10: b9,d9,t are_collinear and
A11: c,d,u are_collinear and
A12: o<>a9 and
A13: d<>d9 and
A14: o<>b9;
A15: d,b,c,b are_coplanar by Th14;
A16: not d,b,c,a are_coplanar by A2,Th7;
  then
A17: not d,b,c are_collinear by Th6;
  not b,d,o are_collinear by A3,Th6;
  then not o,b,d are_collinear by Th1;
  then
A18: t<>d by A4,A6,A10,A13,A14,Th5;
  d,b,t are_collinear & d,c,u are_collinear by A9,A11,Th1;
  then
A19: t<>u by A17,A18,Lm1;
A20: d,b,c,d are_coplanar by Th14;
  not d,o,a are_collinear by A3,Th6;
  then not o,a,d are_collinear by Th1;
  then s<>d by A4,A5,A8,A12,A13,Th5;
  then
A21: not d,b,c,s are_coplanar by A7,A16,Th15;
  b<>d by A2,Th14;
  then
A22: d,b,c,t are_coplanar by A9,A15,A20,Th10;
  d,b,c,c are_coplanar by Th14;
  then d,b,c,u are_coplanar by A1,A3,A11,A20,Th10;
  then not t,u,s are_collinear by A21,A22,A19,Th10;
  hence thesis by Th1;
end;
