reserve X,Y,Z for set, x,y,z for object;
reserve i,j for Nat;
reserve A,B,C for Subset of X;
reserve R,R1,R2 for Relation of X;
reserve AX for Subset of [:X,X:];
reserve SFXX for Subset-Family of [:X,X:];
reserve EqR,EqR1,EqR2,EqR3 for Equivalence_Relation of X;
reserve X for non empty set,
  x for Element of X;
reserve F for Part-Family of X;
reserve e,u,v for object, E,X,Y,X1 for set;

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;
