
theorem Th13:
  for I be non empty set, G be Group,
      F be Group-Family of I
  st for i be Element of I holds F.i is Subgroup of G
  holds 1_product F = I --> 1_G
  proof
    let I be non empty set, G be Group,
        F be Group-Family of I;
    assume
    A1: for i be Element of I holds F.i is Subgroup of G;
    A2: dom(1_product F) = I by Th3;
    A3: dom(I --> 1_G) = I by FUNCOP_1:13;
    for j be object st j in I holds (1_product F).j = (I --> 1_G).j
    proof
      let j be object;
      assume
      A4: j in I; then
      reconsider Z = F.j as Group by Th1;
      A5: (1_product F).j = 1_Z by A4,GROUP_7:6;
      A6: (I --> 1_G).j = 1_G by A4,FUNCOP_1:7;
      reconsider j as Element of I by A4;
      F.j is Subgroup of G by A1;
      hence thesis by A5,A6,GROUP_2:44;
    end;
    hence thesis by A2,A3,FUNCT_1:2;
  end;
