theorem
  for X being set, Y being non empty set, f being Function of X,Y
  for H being Subset-Family of X holds union(.:f.:H) = f.: union H
proof
  let X be set, Y be non empty set, f be Function of X,Y;
  let H be Subset-Family of X;
  dom f = X by FUNCT_2:def 1;
  hence thesis by FUNCT_3:14;
end;
