
theorem Th17:
  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)
  holds support(x,F) c= {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
    A1: x = 1_product F +* (i,g);
    for j be object holds j in support(x,F) implies j in {i}
    proof
      let j be object;
      assume j in support(x,F); then
      ex Z being Group st Z = F.j & x.j <> 1_Z & j in I by Def1; then
      j = i by A1,GROUP_12:1;
      hence thesis by TARSKI:def 1;
    end;
    hence thesis;
  end;
