
theorem Th21:
  for T being Hausdorff non empty TopSpace, S being irreducible
  Subset of T holds S is trivial
proof
  let T be Hausdorff non empty TopSpace, S be irreducible Subset of T;
  assume S is non trivial;
  then consider x,y being Point of T such that
A1: x in S & y in S and
A2: x <> y by SUBSET_1:45;
  consider W,V being Subset of T such that
A3: W is open & V is open and
A4: x in W & y in V and
A5: W misses V by A2,PRE_TOPC:def 10;
  set S1 = S \ W, S2 = S \ V;
A6: S1 <> S & S2 <> S by A4,A1,XBOOLE_0:def 5;
  S is closed by Def3;
  then
A7: S1 is closed & S2 is closed by A3,Th20;
A8: W /\ V = {} by A5;
  S1 \/ S2 = S \ W /\ V by XBOOLE_1:54
    .= S by A8;
  hence contradiction by A7,A6,Def3;
end;
