
theorem Th43:
  for I2 be non empty set, F2 be Group-Family of I2
  st for i be Element of I2 holds card (F2.i) = 1
  holds card (the carrier of sum F2) = 1
  proof
    let I2 be non empty set,
        F2 be Group-Family of I2;
    assume
    A1: for i be Element of I2 holds card (F2.i) = 1;
    A2: for x be object holds 1_ sum F2 = x implies x in [#]sum F2;
    for x be object holds x in [#]sum F2 implies x = 1_ sum F2
    proof
      let x be object;
      assume
      x in [#]sum F2; then
      reconsider x as Element of product F2 by GROUP_2:42;
      dom x = I2 by GROUP_19:3; then
      reconsider x as ManySortedSet of I2 by PARTFUN1:def 2,RELAT_1:def 18;
      for i be set st i in I2
      ex G be Group-like non empty multMagma
      st G = F2.i & x.i = 1_G
      proof
        let i be set;
        assume i in I2; then
        reconsider i as Element of I2;
        reconsider G = F2.i as Group;
        take G;
        A3: card G = 1 by A1; then
        A4: [#] G is finite;
        A5: card {1_G} = 1 by CARD_1:30;
        A6: [#]G = {1_G} by A3,A4,A5,CARD_2:102;
        x in product F2; then
        x.i in F2.i by GROUP_19:5;
        hence thesis by A6,TARSKI:def 1;
      end; then
      x = 1_product F2 by GROUP_7:5;
      hence thesis by GROUP_2:44;
    end; then
    [#]sum F2 = {1_ sum F2} by A2,TARSKI:def 1;
    hence thesis by CARD_1:30;
  end;
