 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 EltGF:
  for F being Group-Family of I
  for g being Element of F
  for i being Element of I
  holds g.i is Element of F.i
proof
  let F be Group-Family of I;
  let g be Element of F;
  let i be Element of I;
  g.i is Element of (Carrier F).i by PBOOLE:def 14;
  hence g.i is Element of F.i by Th9;
end;
