
theorem Th23:
  for I be non empty set, G be Group,
      a be Function of I,G,
      i be Element of I, g be Element of G
  holds support(a) is finite implies support(a +* (i,g)) is finite
  proof
    let I be non empty set, G be Group,
        a be Function of I,G,
        i be Element of I, g be Element of G;
    reconsider b = a +* (i,g) as Function of I,G;
    assume
    A1: support(a) is finite;

    for j be object holds
    j in support(b) implies j in support(a) \/ {i}
    proof
      let j be object;
      assume j in support(b); then
      A2: b.j <> 1_G & j in I by Def2;
      per cases;
      suppose
        i = j; then
        j in {i} by TARSKI:def 1;
        hence thesis by XBOOLE_0:def 3;
      end;
      suppose
        i <> j; then
        b.j = a.j by FUNCT_7:32; then
        j in support(a) by A2,Def2;
        hence thesis by XBOOLE_0:def 3;
      end;
    end; then
    support(b) c= support(a) \/ {i};
    hence thesis by A1;
  end;
