
theorem
  for I be non empty set,
      G be Group,
      i be object,
      a be finite-support Function of I,G,
      b be Function of I,G
  st i in support(a) & b = a +* (i,1_G)
  holds card support(b) = card support(a) - 1
  proof
    let I be non empty set,
        G be Group,
        i be object,
        a be finite-support Function of I,G,
        b be Function of I,G;
    assume
    A1: i in support(a) & b = a +* (i,1_G); then
    support(b) = support(a) \ {i} by Th28; then
    card support(b) = card support(a) - card {i}
                      by A1,CARD_2:44,ZFMISC_1:31
                   .= card support(a) - 1 by CARD_2:42;
    hence thesis;
  end;
