 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 Th37:
  for i,j being Element of I st i <> j
  holds (proj (F,j)) * (1ProdHom (F,i)) = 1:(F.i, F.j)
proof
  let i,j be Element of I;
  assume A1: i <> j;
  set U = the carrier of F.i;
  A2: dom 1:(F.i, F.j) = U & dom ((proj (F,j)) * (1ProdHom (F,i))) = U
  proof
    thus dom 1:(F.i, F.j) = U;
    B1: rng (1ProdHom (F,i)) c= the carrier of ProjGroup (F, i)
    by RELAT_1:def 19;
    the carrier of ProjGroup (F,i) c= the carrier of product F
    by GROUP_2:def 5;
    then (1ProdHom (F,i)) is (the carrier of product F)-valued
    by B1, XBOOLE_1:1, RELAT_1:def 19;
    then dom ((proj (F,j)) * (1ProdHom (F,i))) = dom (1ProdHom (F,i))
    by FUNCT_2:123;
    hence thesis by FUNCT_2:def 1;
  end;
  for x being Element of U
  holds ((proj (F,j)) * (1ProdHom (F,i))).x = (1:(F.i, F.j)).x
  proof
    let x be Element of U;
    B1: dom (1ProdHom (F,i)) = U by FUNCT_2:def 1;
    (1ProdHom (F,i)).x in ProjGroup (F, i);
    then (1_(product F) +* (i, x)) in ProjGroup (F, i) by GROUP_12:def 3;
    then B2: (1_(product F) +* (i, x)) in product F by GROUP_2:40;
    ((proj (F,j)) * (1ProdHom (F,i))).x
     = (proj (F,j)).((1ProdHom (F,i)).x) by B1, FUNCT_1:13
    .= (proj (F,j)).(1_(product F) +* (i, x)) by GROUP_12:def 3
    .= (1_(product F) +* (i, x)).j by Def13, B2
    .= (1_(product F)).j by A1,FUNCT_7:32
    .= (1:(F.i, F.j)).x by GROUP_7:6;
    hence ((proj (F,j)) * (1ProdHom (F,i))).x = (1:(F.i, F.j)).x;
  end;
  hence (proj (F,j)) * (1ProdHom (F,i)) = 1:(F.i, F.j) by A2;
end;
