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 Th21:
  not o,a,b are_collinear & not o,b,c are_collinear & not o,a,c
are_collinear & o,a,a9 are_collinear & o,b,b9 are_collinear &
o,c,c9 are_collinear
& a,b,p are_collinear & a9,b9,p are_collinear & a<>a9 & b,c,r are_collinear &
b9,
c9,r are_collinear & a,c,q are_collinear & b<>b9 & a9,c9,q are_collinear
& o<>a9 &
  o<>b9 & o<>c9 implies r,q,p are_collinear
proof
  assume that
A1: not o,a,b are_collinear and
A2: not o,b,c are_collinear and
A3: not o,a,c are_collinear and
A4: o,a,a9 are_collinear & o,b,b9 are_collinear & o,c,c9 are_collinear and
A5: a,b,p are_collinear and
A6: a9,b9,p are_collinear & a<>a9 and
A7: b,c,r are_collinear and
A8: b9,c9,r are_collinear and
A9: a,c,q are_collinear and
A10: b<>b9 & a9,c9,q are_collinear & o<>a9 & o<>b9 & o<>c9;
A11: now
A12: b<>c & b,c,b are_collinear by A2,ANPROJ_2:def 7;
    assume
A13: a,b,c are_collinear;
    then b,c,a are_collinear by Th1;
    then
A14: a,b,r are_collinear by A7,A12,ANPROJ_2:def 8;
A15: c <>a & a,c,a are_collinear by A3,ANPROJ_2:def 7;
    a,c,b are_collinear by A13,Th1;
    then
A16: a,b,q are_collinear by A9,A15,ANPROJ_2:def 8;
    a<>b by A1,ANPROJ_2:def 7;
    hence thesis by A5,A14,A16,ANPROJ_2:def 8;
  end;
A17: not o,c,a are_collinear by A3,Th1;
  now
    assume not a,b,c are_collinear;
    then o,a,b,c constitute_a_quadrangle by A1,A2,A17;
    then p,r,q are_collinear by A4,A5,A6,A7,A8,A9,A10,Lm7;
    hence thesis by Th1;
  end;
  hence thesis by A11;
end;
