
theorem Th9:
  for I be set,
      G be Group,
      H be Group-Family of I,
      x be Function of I,G,
      y be Element of product H
  st x = y
   & for i be object st i in I holds H.i is Subgroup of G
  holds support(x) = support(y,H)
  proof
    let I be set,
        G be Group,
        H be Group-Family of I,
        x be Function of I,G,
        y be Element of product H;
    assume that
    A1: x = y and
    A2: for i be object st i in I holds H.i is Subgroup of G;
    A5: for i be object, Z being Group st i in I & Z = H.i holds 1_Z = 1_G
    proof
      let i be object;
      let Z be Group;
      assume i in I; then
      H.i is Subgroup of G by A2;
      hence thesis by GROUP_2:44;
    end;
    for i be object holds i in support(y,H) iff i in support(x)
    proof
      let i be object;
      A6: i in support(x) iff (x.i <> 1_G & i in I) by Def2;
      hereby
        assume i in support(y,H); then
        ex Z being Group st Z = H.i & y.i <> 1_Z & i in I by Def1;
        hence i in support(x) by A1,A5,A6;
      end;
      assume
A89:  i in support(x); then
A90:  x.i <> 1_G & i in I by Def2;
      reconsider Z =  H.i as Group by A89,Th1;
      1_Z = 1_G by A5,A89;
      hence i in support(y,H) by A1,A90,Def1;
    end;
    hence support(x) = support(y,H) by TARSKI:2;
  end;
