 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 Th59:
  for S being normal Subgroup-Family of F
  holds product S is normal Subgroup of product F
proof
  let S be normal Subgroup-Family of F;
  for g being Element of product F
  holds (product S) |^ g is Subgroup of product S
  proof
    let g be Element of product F;
    for h being Element of product F st h in (product S) |^ g
    holds h in product S
    proof
      let h be Element of product F;
      assume h in (product S) |^ g;
      then consider a being Element of product F such that
      A2: h = a |^ g & a in product S by GROUP_3:58;
      A3: dom h = I by GROUP_19:3;
      for i being Element of I holds h.i in S.i
      proof
        let i be Element of I;
        h.i = (((g") * a) * g).i by A2,GROUP_3:def 2
           .= (((g") * a)/.i) * (g/.i) by GROUP_7:1
           .= (((g")/.i) * (a/.i)) * (g/.i) by GROUP_7:1
           .= (((g/.i)") * (a/.i)) * (g/.i) by GROUP_7:8
           .= (a/.i) |^ (g/.i) by GROUP_3:def 2;
        then h.i in (S.i) |^ (g/.i) by A2, GROUP_19:5, GROUP_3:58;
        then h.i in the multMagma of S.i by GROUP_3:def 13;
        hence thesis;
      end;
      hence h in product S by A3, Th47;
    end;
    hence (product S) |^ g is Subgroup of product S by GROUP_2:58;
  end;
  hence thesis by GROUP_3:122;
end;
