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 Th18:
  not a,b,c,d are_coplanar & a,b,c,o are_coplanar & not a,b,o
  are_collinear implies not a,b,d,o are_coplanar
proof
  assume that
A1: not a,b,c,d are_coplanar and
A2: a,b,c,o are_coplanar and
A3: not a,b,o are_collinear;
  assume not thesis;
  then
A4: a,b,o,d are_coplanar by Th7;
  a,b,o,c are_coplanar by A2,Th7;
  hence contradiction by A1,A3,A4,Lm4;
end;
