
theorem Th6:
  for T being non empty RelStr, x being Element of T,
  A being lower Subset of T st x in A holds downarrow x c= A
proof
  let T be non empty RelStr, x be Element of T, A be lower Subset of T;
  assume x in A;
  then not x in A` by XBOOLE_0:def 5;
  then A` misses downarrow x by Th5;
  then A` c= (downarrow x)` by SUBSET_1:23;
  hence thesis by SUBSET_1:12;
end;
