 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 Th32:
  for g being Element of product F
  holds proj (F, i) . g = proj (Carrier F, i) . g
proof
  let g be Element of product F;
  set X = product (Carrier F);
  set f = proj (Carrier F, i);
  A1: dom f = X by CARD_3:def 16;
  g in product F;
  then g in dom f by A1, GROUP_7:def 2;
  then (proj (Carrier F, i)) . g = g.i by CARD_3:def 16;
  hence (proj (F, i)) . g = (proj (Carrier F, i)) . g by Def13;
end;
