
theorem Th41:
  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 & A <> Top InclPoset the
  topology of T holds A is irreducible iff B` 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 and
A2: A <> Top InclPoset the topology of T;
A3: the carrier of InclPoset the topology of T = the topology of T
    by YELLOW_1:1;
  hereby
    assume
A4: A is irreducible;
    thus B` is irreducible
    proof
      B <> the carrier of T by A1,A2,YELLOW_1:24;
      then (the carrier of T) \ B <> {} by XBOOLE_1:37;
      hence B` is non empty;
      B`` is open by A1,A3;
      hence B` is closed;
      let S1, S2 be Subset of T such that
A5:   S1 is closed & S2 is closed and
A6:   B` = S1 \/ S2;
A7:   S1` is open & S2` is open by A5;
      then reconsider
      s1 = S1`, s2 = S2` as Element of InclPoset the topology of T
      by A3;
      (S1`) /\ S2` is open by A7;
      then
A8:   s1 /\ s2 in the topology of T;
      B = (S1 \/ S2)` by A6
        .= (S1`) /\ S2` by XBOOLE_1:53;
      then A = s1 "/\" s2 by A1,A8,YELLOW_1:9;
      then A = s1 or A = s2 by A4;
      hence thesis by A1;
    end;
  end;
  thus thesis by A1,Th40;
end;
