reserve x for set;

theorem Th23:
  for T1 being TopSpace, T2 being TopExtension of T1 for A being
  Subset of T1 holds (A is open implies A is open Subset of T2) & (A is closed
  implies A is closed Subset of T2)
proof
  let T1 be TopSpace, T2 be TopExtension of T1;
  let A be Subset of T1;
A1: the carrier of T1 = the carrier of T2 by YELLOW_9:def 5;
  reconsider B = A as Subset of T2 by YELLOW_9:def 5;
  reconsider C = [#]T2 \ B as Subset of T2;
A2: the topology of T1 c= the topology of T2 by YELLOW_9:def 5;
  thus A is open implies A is open Subset of T2
  proof
    assume A in the topology of T1;
    then A in the topology of T2 by A2;
    hence thesis by PRE_TOPC:def 2;
  end;
  assume [#]T1 \ A in the topology of T1;
  then C is open by A2,A1;
  hence thesis by PRE_TOPC:def 3;
end;
