reserve T for non empty TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T;

theorem Th23:
  F is open & G is open implies INTERSECTION (F,G) is open
proof
  assume
A1: F is open & G is open;
  for A being Subset of T st A in INTERSECTION (F,G) holds A is open
  proof
    let A be Subset of T;
    assume A in INTERSECTION (F,G);
    then consider X, Y being set such that
A2: X in F & Y in G and
A3: A = X /\ Y by SETFAM_1:def 5;
    reconsider X, Y as Subset of T by A2;
    X is open & Y is open by A1,A2,TOPS_2:def 1;
    hence thesis by A3;
  end;
  hence thesis by TOPS_2:def 1;
end;
