
theorem Th8:
  for x,y being Element of REAL holds real_dist.(x,y) = 0 iff x = y
proof
  let x,y be Element of REAL;
  thus real_dist.(x,y)=0 implies x=y
  proof
    assume real_dist.(x,y)=0;
    then 0=|.x-y.| by Def12;
    then x-y = 0 by ABSVALUE:2;
    hence thesis;
  end;
  assume x=y;
  then |.x-y.|=0 by ABSVALUE:2;
  hence thesis by Def12;
end;
