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 Th6:
  a,b,c are_collinear or b,c,d are_collinear or c,d,a are_collinear
  or d,a,b are_collinear implies a,b,c,d are_coplanar
proof
A1: now
A2: c,d,c are_collinear by ANPROJ_2:def 7;
    assume a,b,c are_collinear;
    hence thesis by A2;
  end;
A3: now
A4: a,b,a are_collinear by ANPROJ_2:def 7;
    assume c,d,a are_collinear;
    hence thesis by A4;
  end;
A5: now
    assume d,a,b are_collinear;
    then
A6: a,b,d are_collinear by Th1;
    c,d,d are_collinear by ANPROJ_2:def 7;
    hence thesis by A6;
  end;
A7: now
    assume b,c,d are_collinear;
    then
A8: c,d,b are_collinear by Th1;
    a,b,b are_collinear by ANPROJ_2:def 7;
    hence thesis by A8;
  end;
  assume a,b,c are_collinear or b,c,d are_collinear or c,d,a are_collinear
or d,
  a,b are_collinear;
  hence thesis by A1,A7,A3,A5;
end;
