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 Th8:
  c3 <> c1 & c4 <> c1 & c4 <> c3 &
  c3 <> c2 & c4 <> c2 & c6 <> c1 &
  c7 <> c1 & c7 <> c6 & c6 <> c5 &
  c7 <> c5 &
  not c1,c2,c5 are_collinear &
  c1,c2,c3 are_collinear & c1,c2,c4 are_collinear &
  c1,c5,c6 are_collinear & c1,c5,c7 are_collinear implies
  not c1,c3,c6 are_collinear &
  c1,c3,c4 are_collinear & c1,c6,c7 are_collinear &
  c3 <> c1 &  c2 <> c1 &  c3 <> c2 &  c4 <> c3 & c4 <> c2 &
  c6 <> c1 &  c5 <> c1 &  c6 <> c5 &  c7 <> c6 &  c7 <> c5 &
  not c1,c4,c7 are_collinear &
  c1,c4,c3 are_collinear & c1,c4,c2 are_collinear &
  c1,c7,c6 are_collinear & c1,c7,c5 are_collinear
  proof
    assume that
A1: not c3=c1 and
A2: not c4=c1 and
A3: not c4=c3 and
A4: not c3=c2 and
A5: not c4=c2 and
A6: not c6=c1 and
A7: not c7=c1 and
A8: not c7=c6 and
A9: not c6=c5 and
A10: not c7=c5 and
A11: not c1,c2,c5 are_collinear and
A12: c1,c2,c3 are_collinear and
A13: c1,c2,c4 are_collinear and
A14: c1,c5,c6 are_collinear and
A15: c1,c5,c7 are_collinear and
A16: (c1,c3,c6 are_collinear or  not c1,c3,c4 are_collinear or
    not c1,c6,c7 are_collinear or c3=c1 or c2=c1 or c3=c2 or c4=c3 or
    c4=c2 or c6=c1 or c5=c1 or c6=c5 or c7=c6 or c7=c5 or c1,c4,c7
    are_collinear or  not c1,c4,c3 are_collinear or  not c1,c4,c2
    are_collinear or  not c1,c7,c6 are_collinear or  not c1,c7,c5
    are_collinear);
A17: 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 be Element of PCPP;
      v103,v100,v103 are_collinear by COLLSP:5;
      hence thesis by COLLSP:3;
    end;
A18: not c5=c1 by COLLSP:2,A11;
A19: not c2=c1 by COLLSP:2,A11;
A20: c3,c1,c2 are_collinear by A12,HESSENBE:1;
A21: c4,c1,c2 are_collinear by A13,HESSENBE:1;
A22: c6,c1,c5 are_collinear by A14,HESSENBE:1;
A23: c7,c1,c5 are_collinear by A15,HESSENBE:1;
A24: c3,c1,c1 are_collinear by COLLSP:2;
A25: not c3,c1,c5 are_collinear by A24,A20,COLLSP:3,A11,A1;
    now
      thus c4,c1,c1 are_collinear by COLLSP:2;
      c6,c1,c1 are_collinear by COLLSP:2;
      hence not c6,c1,c3 are_collinear by A22,A6,COLLSP:3,A25;
      c7,c1,c1 are_collinear by COLLSP:2;
      hence not c7,c1,c4 are_collinear or c4,c1,c5 are_collinear
        by A23,COLLSP:3,A7;
      thus c1,c3,c3 are_collinear by COLLSP:2;
      thus c1,c4,c4 are_collinear by COLLSP:2;
    end;
    hence contradiction by A13,COLLSP:4,COLLSP:2,A10,
      A8,A9,A5,A3,A4,A16,A15,A17,A1,A19,
      A12,A18,A14,HESSENBE:2,A21,A2,COLLSP:3,A11;
  end;
