
theorem
  for I be non empty set, F1,F2 be Group-Family of I
  st for i be Element of I holds F1.i is Subgroup of F2.i
  holds sum F1 is Subgroup of sum F2
  proof
    let I be non empty set, F1,F2 be Group-Family of I;
    assume
    A1: for i be Element of I holds F1.i is Subgroup of F2.i;
    for x be object st x in [#] sum F1 holds x in [#] sum F2
    proof
      let x be object;
      assume
      A2: x in [#] sum F1; then
      x in sum F1; then
      x in product F1 by GROUP_2:40; then
      reconsider x as Element of product F1;
      product F1 is Subgroup of product F2 by A1,Th20; then
      reconsider y = x as Element of product F2 by GROUP_2:42;
      for i be object holds i in support(y,F2) implies i in support(x,F1)
      proof
        let i be object;
        assume i in support(y,F2); then
        consider Z being Group such that
        A4: Z = F2.i & y.i <> 1_Z & i in I by GROUP_19:def 1;
        reconsider i as Element of I by A4;
        F1.i is Subgroup of F2.i by A1; then
        x.i <> 1_F1.i & i in I by A4,GROUP_2:44;
        hence thesis by GROUP_19:def 1;
      end; then
      support(y,F2) c= support(x,F1); then
      y in sum F2 by A2,GROUP_19:8;
      hence thesis;
    end; then
    A5: [#] sum F1 c= [#] sum F2;
    set mp1 = the multF of product F1;
    set mp2 = the multF of product F2;
    set ms1 = the multF of sum F1;
    set ms2 = the multF of sum F2;
    set cp1 = [#]product F1;
    set cp2 = [#]product F2;
    set cs1 = [#]sum F1;
    set cs2 = [#]sum F2;
    cs1 c= cp1 by GROUP_2:def 5; then
    A6: [:cs1,cs1:] c= [:cp1,cp1:] by ZFMISC_1:96;
    A7: [:cs1,cs1:] c= [:cs2,cs2:] by A5,ZFMISC_1:96;
    A8: product F1 is Subgroup of product F2 by A1,Th20;
    ms1 = mp1 || cs1 by GROUP_2:def 5
       .= (mp2 || cp1) || cs1 by A8,GROUP_2:def 5
       .= mp2 || cs1 by A6,FUNCT_1:51
       .= (mp2 || cs2) || cs1 by A7,FUNCT_1:51
       .= ms2 || cs1 by GROUP_2:def 5;
    hence thesis by A5,GROUP_2:def 5;
  end;
