
theorem Th7:
  for I be set,
      F be Group-Family of I,
      a be Element of sum F holds
  ex J being finite Subset of I,
     b being ManySortedSet of J
  st J = support(a,F)
  & a = 1_product F +* b
  & (for j being object, G being Group st j in I \ J & G = F.j holds a.j = 1_G)
  & (for j being object st j in J holds a.j = b.j)
  proof
    let I be set,
        F be Group-Family of I,
        a be Element of sum F;
    consider g being Element of product Carrier F,
             J being finite Subset of I,
             b being ManySortedSet of J such that
    A1:   g = 1_product F & a = g +* b
        & for j being set st j in J
          ex G being Group-like non empty multMagma
          st G = F.j & b.j in the carrier of G & b.j <> 1_G
          by GROUP_7:def 9;
    take J,b;
A2: dom b = J by PARTFUN1:def 2;
A3: dom(1_product F) = I by Th3;
A4: for j being object, G being Group st j in I \ J & G = F.j holds a.j = 1_G
    proof
      let j be object;
      let G be Group;
      assume that
A6:   j in I \ J and
A7:   G = F.j;
      j in dom(1_product F) & not j in dom b by A3,A6,XBOOLE_0:def 5;
      hence a.j = (1_product F).j by A1,FUNCT_4:11
               .= 1_G by A6,A7,GROUP_7:6;
    end;
    for j be object holds j in support(a,F) iff j in J
    proof
      let j be object;
      hereby
        assume
        A11: j in support(a,F); then
        consider G being Group such that
        A12: G = F.j & a.j <> 1_G & j in I by Def1;
        assume not j in J; then
        j in I \ J by A11,XBOOLE_0:def 5;
        hence contradiction by A4,A12;
      end;
      assume
      A13: j in J; then
      consider G being Group-like non empty multMagma such that
A14:  G = F.j & b.j in the carrier of G & b.j <> 1_G by A1;
      a.j <> 1_G by A1,A2,A13,A14,FUNCT_4:13;
      hence j in support(a,F) by A13,A14,Def1;
    end;
    hence thesis by A1,A2,A4,FUNCT_4:13,TARSKI:2;
  end;
