
theorem Th24:
  for I be non empty set,
      F be Group-Family of I,
      x be Function,
      i be Element of I,
      g be Element of F.i
  holds x in product F implies x +* (i,g) in product F
  proof
    let I be non empty set,
        F be Group-Family of I,
        x be Function,
        i be Element of I,
        g be Element of F.i;
    assume
    A1: x in product F; then
    A2: x in product Carrier F by GROUP_7:def 2; then
    A3: dom x = dom(Carrier F) &
        for j be object st j in dom(Carrier F) holds x.j in (Carrier F).j
        by CARD_3:9;
    A4: dom x = I by A1,Th3;
    set y = x +* (i,g);
    A6: dom y = I by A4,FUNCT_7:30;
    A7: for j be object st j in dom(Carrier F) holds y.j in (Carrier F).j
    proof
      let j be object;
      assume
      A8: j in dom(Carrier F);
      per cases;
      suppose
        A9: j = i; then
        A10: y.j = g by A4,FUNCT_7:31;
        g in [#](F.i);
        hence thesis by A9,A10,PENCIL_3:7;
      end;
      suppose
        j <> i; then
        y.j = x.j by FUNCT_7:32;
        hence thesis by A2,A8,CARD_3:9;
      end;
    end;
    y in product Carrier F by A3,A4,A6,A7,CARD_3:9;
    hence thesis by GROUP_7:def 2;
  end;
