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 Th7:
  c3 <> c1 & c3 <> c2 & c6 <> c1 & c6 <> c5 &
  not c1,c2,c5 are_collinear &
  c1,c2,c3 are_collinear & c1,c5,c6 are_collinear
  implies not c2,c3,c5 are_collinear & not c2,c3,c6 are_collinear &
  not c2,c5,c6 are_collinear & not c3,c5,c6 are_collinear
  proof
    assume that
A1: not c3=c1 and
A2: not c3=c2 and
A3: not c6=c1 and
A4: not c6=c5 and
A5: not c1,c2,c5 are_collinear and
A6: c1,c2,c3 are_collinear and
A7: c1,c5,c6 are_collinear and
A8: c2,c3,c5 are_collinear or c2,c3,c6 are_collinear or
       c2,c5,c6 are_collinear or c3,c5,c6 are_collinear;
A9: not c2,c5,c1 are_collinear by HESSENBE:1,A5;
A10: c2,c3,c1 are_collinear by A6,COLLSP:8;
A11: c5,c6,c1 are_collinear by A7,COLLSP:8;
    now
      thus c2,c3,c6 are_collinear or c2,c5,c6 are_collinear or
        c5,c6,c3 are_collinear by A2,HESSENBE:2,A10,A9,A8,COLLSP:8;
      c6,c1,c1 are_collinear & c6,c1,c5 are_collinear
        by COLLSP:2,A7,HESSENBE:1;
      hence not c6,c1,c3 are_collinear or c3,c1,c5 are_collinear
        by A3,COLLSP:3;
      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:2;
        hence thesis by COLLSP:3;
      end;
      hence not c5,c6,c2 are_collinear by A11,A4,A9;
      thus c2,c3,c3 are_collinear by COLLSP:2;
      thus c2,c5,c5 are_collinear by COLLSP:2;
      c3,c1,c1 are_collinear & c3,c1,c2 are_collinear
        by COLLSP:2,A6,HESSENBE:1;
      hence not c3,c1,c5 are_collinear by A1,COLLSP:3,A5;
      thus not c5=c2 by COLLSP:2,A5;
      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 be Element of PCPP;
        v104,v100,v104 are_collinear by COLLSP:2;
        hence thesis by COLLSP:3;
      end;
    end;
    hence contradiction by A2,COLLSP:3,A10,A11,A4;
  end;
