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 Th34:
  union(F|M) c= union F
proof
  let x be object;
  assume x in union(F|M);
  then consider A being set such that
A1: x in A and
A2: A in F|M by TARSKI:def 4;
  reconsider Q = A as Subset of T|M by A2;
  consider R being Subset of T such that
A3: R in F and
A4: R /\ M = Q by A2,Def3;
  x in R by A1,A4,XBOOLE_0:def 4;
  hence thesis by A3,TARSKI:def 4;
end;
