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 Th10:
  p<>q & p,q,r are_collinear & a,b,c,p are_coplanar & a,b,c,q
  are_coplanar implies a,b,c,r are_coplanar
proof
  assume that
A1: p<>q and
A2: p,q,r are_collinear and
A3: a,b,c,p are_coplanar and
A4: a,b,c,q are_coplanar;
A5: q,p,r are_collinear by A2,Th1;
  now
    assume
A6: not a,b,c are_collinear;
    then a,b,p,q are_coplanar by A3,A4,Lm4;
    then
A7: a,b,p,r are_coplanar by A1,A2,Lm6;
A8: now
      b,a,b are_collinear by ANPROJ_2:def 7;
      then
A9:   a,b,p,b are_coplanar by Th6;
      a,a,b are_collinear by ANPROJ_2:def 7;
      then
A10:  a,b,p,a are_coplanar by Th6;
      assume
A11:  not a,b,p are_collinear;
      a,b,p,c are_coplanar by A3,Th7;
      hence thesis by A7,A11,A10,A9,Th8;
    end;
    a,b,q,p are_coplanar by A3,A4,A6,Lm4;
    then
A12: a,b,q,r are_coplanar by A1,A5,Lm6;
A13: now
      b,a,b are_collinear by ANPROJ_2:def 7;
      then
A14:  a,b,q,b are_coplanar by Th6;
      a,a,b are_collinear by ANPROJ_2:def 7;
      then
A15:  a,b,q,a are_coplanar by Th6;
      assume
A16:  not a,b,q are_collinear;
      a,b,q,c are_coplanar by A4,Th7;
      hence thesis by A12,A16,A15,A14,Th8;
    end;
    now
      assume a,b,p are_collinear & a,b,q are_collinear;
      then a,b,r are_collinear by A1,A2,COLLSP:9;
      then r,a,b are_collinear by Th1;
      hence thesis by Th6;
    end;
    hence thesis by A8,A13;
  end;
  hence thesis by Th6;
end;
