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

theorem Th22:
  F is closed & G is closed implies UNION (F,G) is closed
proof
  assume
A1: F is closed & G is closed;
  for A being Subset of T st A in UNION (F,G) holds A is closed
  proof
    let A be Subset of T;
    assume A in UNION (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 4;
    reconsider X, Y as Subset of T by A2;
    X is closed & Y is closed by A1,A2,TOPS_2:def 2;
    hence thesis by A3;
  end;
  hence thesis by TOPS_2:def 2;
end;
