
theorem Th44:
  for I be non empty set,
      G be Group,
      x be finite-support Function of I,G
  st for i be object st i in I holds x.i = 1_G
  holds Product x = 1_G
  proof
    let I be non empty set,
        G be Group,
        x be finite-support Function of I,G;
    assume
    A1: for i be object st i in I holds x.i = 1_G;
    support(x) = {}
    proof
      assume not support(x) = {}; then
      consider i be object such that
      A2: i in support(x) by XBOOLE_0:def 1;
      x.i <> 1_G & i in I by A2,GROUP_19:def 2;
      hence contradiction by A1;
    end;
    hence thesis by GROUP_19:15;
  end;
