 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 Th38:
  (proj (F,i)) * (1ProdHom (F, i)) = id (the carrier of F.i)
proof
  set U = the carrier of F.i;
  A1: (1ProdHom (F, i)) is Homomorphism of F.i, product F by GROUP_19:6;
  for x being Element of U
  holds ((proj (F,i)) * (1ProdHom (F, i))).x = x
  proof
    let x be Element of U;
    B1: dom (1_(product F)) = I by GROUP_19:3;
    B2: (1ProdHom (F, i)).x = (1_(product F)) +* (i, x) by GROUP_12:def 3;
    (1ProdHom (F, i)).x in (ProjGroup (F,i));
    then (1ProdHom (F, i)).x in product F by GROUP_2:40;
    then B3: (proj (F, i)).((1_(product F)) +* (i, x))
    = ((1_(product F)) +* (i, x)).i by B2, Def13;
    dom (1ProdHom (F, i)) = U by FUNCT_2:def 1;
    then ((proj (F,i)) * (1ProdHom (F, i))).x
     = (proj (F, i)).((1ProdHom (F, i)).x) by FUNCT_1:13
    .= (proj (F, i)).((1_(product F)) +* (i, x)) by GROUP_12:def 3
    .= x by B1, B3, FUNCT_7:31;
    hence thesis;
  end;
  hence thesis by A1, FUNCT_2:124;
end;
