
theorem Th40:
  for T being non empty TopSpace for A being Element of InclPoset
the topology of T for B being Subset of T st A = B & B` is irreducible holds A
  is irreducible
proof
  let T be non empty TopSpace, A be Element of InclPoset the topology of T, B
  be Subset of T such that
A1: A = B;
  assume that
  B` is non empty closed and
A2: for S1, S2 being Subset of T st S1 is closed & S2 is closed & B` =
  S1 \/ S2 holds S1 = B` or S2 = B`;
  let x, y be Element of InclPoset the topology of T such that
A3: A = x "/\" y;
A4: the carrier of InclPoset the topology of T = the topology of T by
YELLOW_1:1;
  then x in the topology of T & y in the topology of T;
  then reconsider x1 = x, y1 = y as Subset of T;
A5: y1 is open by A4;
  then
A6: y1` is closed;
A7: x1 is open by A4;
  then x1 /\ y1 is open by A5;
  then x /\ y in the topology of T;
  then A = x /\ y by A3,YELLOW_1:9;
  then
A8: B` = (x1`) \/ y1` by A1,XBOOLE_1:54;
  x1` is closed by A7;
  then x1` = B` or y1` = B` by A2,A6,A8;
  hence thesis by A1,TOPS_1:1;
end;
