
theorem
  for I be non empty set,
      F be Group-Family of I,
      x be Element of product F,
      i be Element of I,
      g be Element of F.i
  st x = 1_product F +* (i,g)
   & g <> 1_F.i
  holds support(x,F) = {i}
  proof
    let I be non empty set,
        F be Group-Family of I,
        x be Element of product F,
        i be Element of I,
        g be Element of F.i;
    assume that
    A1: x = 1_product F +* (i,g) and
    A2: g <> 1_F.i;
    A3: dom x = I & x.i = g
      & for j be Element of I st j <> i holds x.j = 1_F.j by A1,GROUP_12:1;
    A4: support(x,F) c= {i} by A1,Th17;
    i in support(x,F) by A2,A3,Def1; then
    {i} c= support(x,F) by ZFMISC_1:31;
    hence thesis by A4,XBOOLE_0:def 10;
  end;
