 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 i being Element of I
  for g being Element of product F
  holds (g +* (i, 1_(F.i))) in Ker (proj (F, i))
proof
  let i be Element of I;
  let g be Element of product F;
  A1: dom g = I by GROUP_19:3;
  (g +* (i, 1_(F.i))) in product F by Th34;
  then reconsider h=(g +* (i, 1_(F.i))) as Element of product F;
  (proj (F, i)).h = h.i by Def13
                 .= 1_(F.i) by A1, FUNCT_7:31;
  hence (g +* (i, 1_(F.i))) in Ker (proj (F, i)) by GROUP_6:41;
end;
