 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
  for G being Group
  for I being empty set
  for F being Subgroup-Family of I,G
  holds G is_internal_product_of F iff G is trivial
proof
  let G be Group;
  let I be empty set;
  let F be Subgroup-Family of I,G;
  thus G is_internal_product_of F implies G is trivial
    by ThJoinEmptyGr;
  assume G is trivial;
  then the multMagma of G = (1).G by GROUP_22:6;
  hence G is_internal_product_of F by ThJoinEmptyGr;
end;
