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 Th14:
  c3 <> c1 &
  not c1,c2,c5 are_collinear &
  c1,c2,c3 are_collinear & c1,c5,c7 are_collinear implies
  c7 <> c3
  proof
    assume that
A1: not c3=c1 and
A2: not c1,c2,c5 are_collinear and
A3: c1,c2,c3 are_collinear and
A4: c1,c5,c7 are_collinear and
A5: c7=c3;
    c3,c1,c2 are_collinear & c3,c1,c5 are_collinear & c3,c1,c1 are_collinear
      by A3,A4,A5,HESSENBE:1,COLLSP:2;
    hence contradiction by A1,COLLSP:3,A2;
  end;
