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 Th41:
  for X,Y being 1-sorted, f being Function of X, Y st [#]Y = {}
  implies [#]X = {} holds f"([#]Y) = [#]X
proof
  let X,Y be 1-sorted, f be Function of X, Y such that
A1: [#]Y={} implies [#]X={};
  f"(rng f) = dom f by RELAT_1:134
    .= [#]X by A1,FUNCT_2:def 1;
  then [#]X c= f"([#]Y) by RELAT_1:143;
  hence thesis by XBOOLE_0:def 10;
end;
