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 Th5:
  not c2=c1 & not c3=c2 &
  not c5=c1 & not c7=c5 &
  not c7=c6 & not c1,c4,c7 are_collinear &
  c1,c4,c2 are_collinear & c1,c4,c3 are_collinear &
  c1,c7,c5 are_collinear & c1,c7,c6 are_collinear &
  c4,c5,c8 are_collinear & c7,c2,c8 are_collinear &
  c2,c6,c10 are_collinear & c3,c5,c10 are_collinear implies
  not c10,c7,c8 are_collinear
  proof
    assume that
A1: not c2=c1 and
A2: not c3=c2 and
A3: not c5=c1 and
A4: not c7=c5 and
A5: not c7=c6 and
A6: not c1,c4,c7 are_collinear and
A7: c1,c4,c2 are_collinear and
A8: c1,c4,c3 are_collinear and
A9: c1,c7,c5 are_collinear and
A10: c1,c7,c6 are_collinear and
A11: c4,c5,c8 are_collinear and
A12: c7,c2,c8 are_collinear and
A13: c2,c6,c10 are_collinear and
A14: c3,c5,c10 are_collinear and
A15: c10,c7,c8 are_collinear;
A16: not c4=c1 by COLLSP:2,A6;
    now
      now
        c1,c4,c4 are_collinear by COLLSP:2;
        hence c3,c2,c4 are_collinear by A8,A7,A16,COLLSP:3;
        thus not c1,c3,c2 are_collinear or c3,c2,c1 are_collinear
          by HESSENBE:1;
        c1,c5,c1 are_collinear & c1,c5,c7 are_collinear
          by COLLSP:2,A9,HESSENBE:1;
        hence not c1,c5,c4 are_collinear by A3,COLLSP:3,A6;
        thus c1,c4,c1 are_collinear by COLLSP:2;
        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:2;
          hence thesis by COLLSP:3;
        end;
        thus c7,c8,c10 are_collinear by A15,HESSENBE:1;
        thus c2,c10,c6 are_collinear by A13,HESSENBE:1;
      end;
      hence not c3,c2,c5 are_collinear & (c8=c7 or
        not c7,c8,c2 are_collinear or c10,c2,c7 are_collinear) &
        (c10=c2 or not c2,c10,c7 are_collinear or c6,c7,c2 are_collinear)
        by A7,A16,A8,A2,COLLSP:3;
      now
        thus c7,c6,c7 are_collinear by COLLSP:2;
        c1,c2,c1 are_collinear & c1,c2,c4 are_collinear
          by COLLSP:2,A7,HESSENBE:1;
        hence not c1,c2,c7 are_collinear by A1,COLLSP:3,A6;
        thus c7,c6,c1 are_collinear by A10,HESSENBE:1;
      end;
      hence not c7,c6,c2 are_collinear by A5,COLLSP:3;
      c7,c5,c7 are_collinear & c7,c5,c1 are_collinear
        by COLLSP:2,A9,HESSENBE:1;
      hence not c7,c5,c4 are_collinear by COLLSP:3,A4,A6;
    end;
    hence contradiction by A11,HESSENBE:1,A12,A14;
  end;
