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 Th12:
  not c1,c2,c5 are_collinear &
  c1,c2,c3 are_collinear & c1,c2,c4 are_collinear implies
  c2,c3,c4 are_collinear
  proof
    assume that
A1: not c1,c2,c5 are_collinear and
A2: c1,c2,c3 are_collinear and
A3: c1,c2,c4 are_collinear and
A4: not c2,c3,c4 are_collinear;
    now
      thus for v100,v2 being Element of PCPP holds
        v2,v100,v100 are_collinear by COLLSP:2;
      not c2=c1 or c1,c2,c5 are_collinear by COLLSP:2;
      hence for v0 being Element of PCPP holds  not c1,c2,v0 are_collinear
        or not c1,c2,c4 are_collinear or v0,c3,c4 are_collinear
        by A2,COLLSP:3,A1;
    end;
    hence contradiction by A3,A4;
  end;
