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

theorem
  for F, G being set holds UNION (F, G) = {} implies F = {} or G = {}
proof
  let F, G be set;
  assume
A1: UNION (F,G) = {};
  assume that
A2: F <> {} and
A3: G <> {};
  consider X being object such that
A4: X in F by A2,XBOOLE_0:def 1;
  consider Y being object such that
A5: Y in G by A3,XBOOLE_0:def 1;
  reconsider Y,X as set by TARSKI:1;
  X \/ Y in UNION (F,G) by A4,A5,SETFAM_1:def 4;
  hence thesis by A1;
end;
