
theorem Th23:
  for S, T being non empty TopSpace, A being irreducible Subset of
S, B being Subset of T st A = B & the TopStruct of S = the TopStruct of T holds
  B is irreducible
proof
  let S, T be non empty TopSpace, A be irreducible Subset of S, B be Subset of
  T such that
A1: A = B and
A2: the TopStruct of S = the TopStruct of T;
  A is non empty closed by YELLOW_8:def 3;
  hence B is non empty closed by A1,A2,TOPS_3:79;
  let B1, B2 be Subset of T such that
A3: B1 is closed & B2 is closed and
A4: B = B1 \/ B2;
  reconsider A1 = B1, A2 = B2 as Subset of S by A2;
  A1 is closed & A2 is closed by A2,A3,TOPS_3:79;
  hence thesis by A1,A4,YELLOW_8:def 3;
end;
