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 Th10:
  c4 <> c1 & c6 <> c5 & not c1,c2,c5 are_collinear implies
  not (c1,c2,c4 are_collinear) or not (c1,c5,c6 are_collinear) or
    not (c4,c6,c8 are_collinear) or (c8 <> c5)
  proof
    assume that
A1: not c4=c1 and
A2: not c6=c5 and
A3: not c1,c2,c5 are_collinear and
A4: c1,c2,c4 are_collinear and
A5: c1,c5,c6 are_collinear and
A6: c4,c6,c8 are_collinear and
A7: c8=c5;
     now
       thus not c5,c1,c2 are_collinear by COLLSP:8,A3;
       thus c5,c4,c6 are_collinear by A6,A7,HESSENBE:1;
       c4,c1,c1 are_collinear & c4,c1,c2 are_collinear
         by A4,COLLSP:2,HESSENBE:1;
       hence for v0 being Element of PCPP holds
         not c4,c1,v0 are_collinear or v0,c1,c2 are_collinear
         by A1,COLLSP:3;
       now
         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 c5,c6,c1 are_collinear by A5,COLLSP:8;
       end;
       hence not c5,c6,c4 are_collinear or c4,c1,c5 are_collinear by A2;
     end;
     hence contradiction by COLLSP:4;
   end;
