
theorem Th33:
  for I be non empty set,
      F be Group-Family of I,
      i be Element of I
  holds ProjGroup(F,i) is Subgroup of sum F
  proof
    let I be non empty set,
        F be Group-Family of I,
        i be Element of I;
    set S = ProjGroup(F,i);
    set G = sum F;
    for x be object st x in [#]S holds x in [#]G
    proof
      let x be object;
      assume
      A1: x in [#]S; then
      x in S; then
      x in product F by GROUP_2:40; then
      reconsider x as Element of product F;
      x in ProjSet(F,i) by A1,GROUP_12:def 2; then
      consider h be Element of F.i such that
      A3: x = 1_product F +* (i,h) by GROUP_12:def 1;
      support(x,F) c= {i} by A3,Th17; then
      x in sum F by Th8;
      hence thesis;
    end; then
    A14: [#]S c= [#]G;
    A16: the multF of G
      = (the multF of (product F)) || (the carrier of G) by GROUP_2:def 5
     .= (the multF of (product F)) | [:[#]G,[#]G:] by REALSET1:def 2;
    (the multF of G) || [#]S
      = ((the multF of (product F)) | [:[#]G,[#]G:]) | [:[#]S,[#]S:]
        by A16,REALSET1:def 2
     .= (the multF of (product F)) | [:[#]S,[#]S:]
        by A14,RELAT_1:74,ZFMISC_1:96
     .= (the multF of (product F)) || [#]S by REALSET1:def 2
     .= the multF of S by GROUP_2:def 5;
    hence thesis by A14,GROUP_2:def 5;
  end;
