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;

theorem Th16:
  c4 <> c1 & c4 <> c2 & c6 <> c1 &
  c7 <> c6 & c7 <> c5 &
  not c1,c2,c5 are_collinear &
  c1,c2,c4 are_collinear & c1,c5,c6 are_collinear &
  c1,c5,c7 are_collinear & c2,c7,c9 are_collinear &
  c4,c5,c9 are_collinear implies
  not c9,c2,c5 are_collinear
  proof
    assume that
A1: not c4=c1 and
A2: not c4=c2 and
A3: not c6=c1 and
A4: not c7=c6 and
A5: not c7=c5 and
A6: not c1,c2,c5 are_collinear and
A7: c1,c2,c4 are_collinear and
A8: c1,c5,c6 are_collinear and
A9: c1,c5,c7 are_collinear and
A10: c2,c7,c9 are_collinear and
A11: c4,c5,c9 are_collinear and
A12: c9,c2,c5 are_collinear;
A13: for v102,v103,v100,v104 being Element of PCPP holds v100=v104 or
       not v104,v100,v102 are_collinear or
       not v104,v100,v103 are_collinear or v102,v103,v104 are_collinear
    proof
      let v102,v103,v100,v104 being Element of PCPP;
      v104,v100,v104 are_collinear by COLLSP:2;
      hence thesis by COLLSP:3;
    end;
A14: for v102,v104,v100,v103 being Element of PCPP holds v100=v103 or
       not v103,v100,v102 are_collinear or
       not v103,v100,v104 are_collinear or v102,v103,v104 are_collinear
    proof
      let v102,v104,v100,v103 being Element of PCPP;
      v103,v100,v103 are_collinear by COLLSP:2;
      hence thesis by COLLSP:3;
    end;
A15: not c5=c1 by COLLSP:2,A6;
A16: c5,c7,c1 are_collinear by A9,HESSENBE:1;
    now
      not c1,c6,c7 are_collinear or c6,c7,c1 are_collinear by HESSENBE:1;
      hence not c6,c7,c2 are_collinear or c2,c6,c1 are_collinear
        by A15,A8,HESSENBE:2,A9,A14,A4;
      thus c9=c2 or  not c9,c2,c7 are_collinear or c9,c7,c5 are_collinear
        by A12,HESSENBE:2;
      thus not c5,c7,c9 are_collinear or c9,c5,c1 are_collinear by A5,A16,A14;
      thus not c1,c9,c5 are_collinear or c1,c5,c9 are_collinear by COLLSP:4;
      thus for v0 being Element of PCPP holds  not c1,c5,v0 are_collinear or
        c7,v0,c6 are_collinear by A15,A8,COLLSP:3,A9;
      now
        thus c4,c5,c5 are_collinear by COLLSP:2;
        c4,c1,c1 are_collinear & c4,c1,c2 are_collinear
          by COLLSP:2,A7,HESSENBE:1;
        then not c4,c1,c5 are_collinear by A1,COLLSP:3,A6;
        hence not c5=c4 & not c5,c7,c4 are_collinear by COLLSP:2,A16,A5,A13;
      end;
      hence not c4,c5,c7 are_collinear by A13;
      c7,c9,c2 are_collinear by A10,HESSENBE:1;
      hence for v0 being Element of PCPP holds c9=c7 or
        not c7,c9,v0 are_collinear or v0,c7,c2 are_collinear by A14;
      c6,c1,c1 are_collinear & c6,c1,c5 are_collinear
        by A8,HESSENBE:1,COLLSP:2;
      hence not c6,c1,c2 are_collinear by COLLSP:3,A3,A6;
      not c5,c1,c2 are_collinear & c2,c4,c1 are_collinear
        by A7,HESSENBE:1,A6;
      hence not c2,c4,c5 are_collinear by A2,A13;
    end;
    hence contradiction by A11,A10,HESSENBE:1;
  end;
