reserve T, T1 for non empty TopSpace;
reserve F,G,H for Subset-Family of T,
  A,B,C,D for Subset of T,
  O,U for open Subset of T,
  p,q for Point of T,
  x,y,X for set;
reserve Un for FamilySequence of T,
  r,r1,r2 for Real,
  n for Element of NAT;

theorem Th17:
  for U being sequence of  bool(the carrier of T) st (for n
  holds U.n is open) holds Union U is open
proof
  let U be sequence of  bool(the carrier of T);
  assume
A1: for n holds U.n is open;
  rng U c= the topology of T
  proof
    let u be object;
    assume u in rng U;
    then consider k being object such that
A2: k in dom U and
A3: u = U.k by FUNCT_1:def 3;
    reconsider k as Element of NAT by A2;
    U.k is open by A1;
    hence thesis by A3,PRE_TOPC:def 2;
  end;
  then union(rng U) in the topology of T by PRE_TOPC:def 1;
  then Union U in the topology of T by CARD_3:def 4;
  hence thesis by PRE_TOPC:def 2;
end;
