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 Th9:
  c4 <> c2 & c4 <> c3 & c8 <> c2 &
  not c2,c3,c6 are_collinear implies
  not c2,c3,c4 are_collinear or
  not c2,c6,c8 are_collinear or
  not c3,c4,c8 are_collinear
  proof
    assume that
A1: not c4=c2 and
A2: not c4=c3 and
A3: not c8=c2 and
A4: not c2,c3,c6 are_collinear and
A5: c2,c3,c4 are_collinear and
A6: c2,c6,c8 are_collinear and
A7: c3,c4,c8 are_collinear;
    now
      thus c4,c2,c3 are_collinear by A5,HESSENBE:1;
      thus c3,c4,c2 are_collinear by A5,COLLSP:8;
      thus c4,c2,c2 are_collinear by COLLSP:2;
      c8,c2,c2 are_collinear & c8,c2,c6 are_collinear
        by A6,HESSENBE:1,COLLSP:2;
      hence (not c8,c2,c4 are_collinear or c4,c2,c6 are_collinear) &
        c3,c4,c4 are_collinear by A3,COLLSP:2,3;
    end;
    hence contradiction by A7,A1,A4,A2,COLLSP:3;
  end;
