reserve x1, x2, x3, x4, x5, x6, x7 for set;

theorem
  for A being Subset of R^1 st REAL = A` holds A = {}
proof
  reconsider AB = {}R^1 as Subset of R^1;
  let A be Subset of R^1;
  assume REAL = A`;
  then AB` = A` by TOPMETR:17;
  then AB = A``;
  hence thesis;
end;
