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 Th15:
  c4 <> c1 & c4 <> c3 &
  not c1,c2,c5 are_collinear &
  c1,c2,c3 are_collinear & c1,c2,c4 are_collinear &
  c4,c5,c9 are_collinear implies
  c9 <> c3
  proof
    assume that
A1: not c4=c1 and
A2: not c4=c3 and
A3: not c1,c2,c5 are_collinear and
A4: c1,c2,c3 are_collinear and
A5: c1,c2,c4 are_collinear and
A6: c4,c5,c9 are_collinear and
A7: c9=c3;
    now
      thus not c2=c1 or c1,c2,c5 are_collinear by COLLSP:2;
      thus c1,c4,c2 are_collinear by A5,COLLSP:4;
      now
        thus 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:5;
          hence thesis by COLLSP:3;
        end;
        thus not c4,c5,c3 are_collinear or c4,c3,c5 are_collinear by COLLSP:4;
      end;
      hence not c4,c3,c1 are_collinear or c1,c4,c5 are_collinear
        by A6,A7,A2;
      thus 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;
    end;
    hence contradiction by A1,A3,A4,HESSENBE:2,A5;
  end;
