
theorem
  for I be non empty set,
      F be Group-Family of I,
      i be Element of I,
      a be Element of sum F,
      b be Function
  st i in support(a,F) & b = a +* (i,1_F.i)
  holds card support(b,F) = card support(a,F) - 1
  proof
    let I be non empty set,
        F be Group-Family of I,
        i be Element of I,
        a be Element of sum F,
        b be Function;
    assume that
    A1: i in support(a,F) and
    A2: b = a +* (i,1_F.i);
    a is Element of product F by GROUP_2:42; then
    dom a = I by Th3; then
    support(b,F) = support(a,F) \ {i} by A2,Th27; then
    card support(b,F) = card support(a,F) - card {i}
                        by A1,CARD_2:44,ZFMISC_1:31
                     .= card support(a,F) - 1 by CARD_2:42;
    hence thesis;
  end;
