 reserve I for non empty set;
 reserve i for Element of I;
 reserve F for Group-Family of I;
 reserve G for Group;
reserve S for Subgroup-Family of F;
reserve f for Homomorphism-Family of G, F;

theorem ThJoinEmptyGr:
  for G being Group
  for I being empty set
  for F being Subgroup-Family of I,G
  holds gr Union (Carrier F) = (1).G
proof
  let G be Group;
  let I be empty set;
  let F be Subgroup-Family of I,G;
  Carrier F = {} --> (bool the carrier of G);
  then Union Carrier F = {} by FUNCT_6:26;
  then Union Carrier F = {} the carrier of G by SUBSET_1:def 2;
  hence gr Union (Carrier F) = (1).G by GROUP_4:30;
end;
