 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 S being Group-Family of I
  for g being Element of product F
  holds g in product S iff (for i being Element of I
                            holds (proj (F,i)).g in S.i)
proof
  let S be Group-Family of I;
  let g be Element of product F;
  hereby
    assume A1: g in product S;
    let i be Element of I;
    g.i in S.i by A1, GROUP_19:5;
    hence (proj (F,i)).g in S.i by Def13;
  end;
  assume Z2: for i being Element of I holds (proj (F,i)).g in S.i;
  Z3: dom g = I by GROUP_19:3;
  for i being Element of I holds g.i in S.i
  proof
    let i be Element of I;
    (proj (F, i)).g in S.i by Z2;
    hence g.i in S.i by Def13;
  end;
  hence g in product S by Z3, Th47;
end;
