
theorem Th6:
  for T being transitive lower non empty TopRelStr for A being
  Subset of T st A is closed holds A is upper
proof
  let T be transitive lower non empty TopRelStr;
  let A be Subset of T;
  assume [#]T\A in the topology of T;
  then A` is open;
  then
A1: A` is lower by Th5;
  let x,y be Element of T;
  assume that
A2: x in A and
A3: x <= y and
A4: not y in A;
A5: y in A` by A4,XBOOLE_0:def 5;
  not x in A` by A2,XBOOLE_0:def 5;
  hence thesis by A5,A1,A3;
end;
