reserve AFV for WeakAffVect;
reserve a,b,c,d,e,f,a9,b9,c9,d9,f9,p,q,r,o,x99 for Element of AFV;

theorem
  ex a,b st a<>b & not MDist a,b
proof
  consider p,q such that
A1: p <> q by STRUCT_0:def 10;
  now
    consider r such that
A2: p,r // r,q by Def1;
A3: now
A4:   now
        assume MDist p,r;
        then
A5:     p,r // r,p;
        r,q // p,r by A2,Th3;
        then q,r // r,p by Th7;
        then p,r // q,r by A5,Def1;
        hence thesis by A1,Th4,Th7;
      end;
      assume p <> r;
      hence thesis by A4;
    end;
    now
      assume
A6:   p = r;
      then r,q // p,p by A2,Th3;
      hence thesis by A1,A6,Def1;
    end;
    hence thesis by A3;
  end;
  hence thesis;
end;
