reserve a,b,c,d,e,f,g,h,i for Real,
                        M for Matrix of 3,REAL;
reserve                           PCPP for CollProjectiveSpace,
        c1,c2,c3,c4,c5,c6,c7,c8,c9,c10 for Element of PCPP;
reserve o,p1,p2,p3,q1,q2,q3,r1,r2,r3 for Element of ProjectiveSpace TOP-REAL 3;

theorem
  o<>p2 & o<>p3 & p2<>p3 &
  p1<>p2 & p1<>p3 & o<>q2 & o<>q3 & q2<>q3 & q1<> q2 & q1<>q3 &
  not o,p1,q1 are_collinear & o,p1,p2 are_collinear & o,p1,p3 are_collinear &
  o,q1,q2 are_collinear & o,q1,q3 are_collinear & p1,q2,r3 are_collinear &
  q1,p2,r3 are_collinear & p1,q3,r2 are_collinear & p3,q1,r2 are_collinear &
  p2,q3,r1 are_collinear & p3,q2,r1 are_collinear
  implies r1,r2,r3 are_collinear
  proof
    assume
A1: o<>p2 & o<>p3 & p2<>p3 & p1<>p2 & p1<>p3 & o<>q2 & o<>q3 & q2<>q3 &
      q1<> q2 & q1<>q3 & not o,p1,q1 are_collinear & o,p1,p2 are_collinear &
      o,p1,p3 are_collinear & o,q1,q2 are_collinear & o,q1,q3 are_collinear &
      p1,q2,r3 are_collinear & q1,p2,r3 are_collinear &
      p1,q3,r2 are_collinear &
      p3,q1,r2 are_collinear & p2,q3,r1 are_collinear & p3,q2,r1 are_collinear;
    per cases;
    suppose p1,q1,r1 are_collinear;
      hence thesis by A1,Lm7;
    end;
    suppose not p1,q1,r1 are_collinear;
      hence thesis by A1,Lm5;
    end;
  end;
