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 Th11:
  c4 <> c2 & c6 <> c1 &
  not c1,c2,c5 are_collinear &
  c1,c2,c4 are_collinear & c1,c5,c6 are_collinear &
  c4,c6,c8 are_collinear implies c8 <> c2
  proof
    assume that
A1: not c4=c2 and
A2: not c6=c1 and
A3: not c1,c2,c5 are_collinear and
A4: c1,c2,c4 are_collinear and
A5: c1,c5,c6 are_collinear and
A6: c4,c6,c8 are_collinear and
A7: c8=c2;
    now
      c6,c1,c1 are_collinear & c6,c1,c5 are_collinear
        by COLLSP:2,A5,HESSENBE:1;
      hence not c6,c1,c2 are_collinear by A2,COLLSP:3,A3;
      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:5;
        hence thesis by COLLSP:3;
      end;
      hence for v3,v2 being Element of PCPP holds c4=c2 or
        not c2,c4,v2 are_collinear or not c2,c4,v3 are_collinear or
        v2,v3,c2 are_collinear;
      thus c2,c4,c6 are_collinear by A6,A7,HESSENBE:1;
      thus c2,c4,c1 are_collinear by A4,COLLSP:8;
    end;
    hence contradiction by A1;
  end;
