reserve x, y for set,
  T for TopStruct,
  GX for TopSpace,
  P, Q, M, N for Subset of T,
  F, G for Subset-Family of T,
  W, Z for Subset-Family of GX,
  A for SubSpace of T;

theorem
  for T being 1-sorted, F, G being Subset-Family of T holds union F \
  union G c= union(F \ G)
proof
  let T be 1-sorted, F, G be Subset-Family of T;
  let x be object;
  assume
A1: x in union F \ union G;
  then x in union F by XBOOLE_0:def 5;
  then consider A being set such that
A2: x in A and
A3: A in F by TARSKI:def 4;
  not x in union G by A1,XBOOLE_0:def 5;
  then not A in G by A2,TARSKI:def 4;
  then A in F \ G by A3,XBOOLE_0:def 5;
  hence thesis by A2,TARSKI:def 4;
end;
