
theorem Th25:
  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 sum F implies x +* (i,g) in sum 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;
    set y = x +* (i,g);
    assume
    A1: x in sum F; then
    A2: y in product F by Th24,GROUP_2:40;
    for j be object holds j in support(y,F) implies j in support(x,F) \/ {i}
    proof
      let j be object;
      assume j in support(y,F); then
      consider Z being Group such that
      A3: Z = F.j & y.j <> 1_Z & j in I by Def1;
      per cases;
      suppose
        j = i; then
        j in {i} by TARSKI:def 1;
        hence thesis by TARSKI:def 3,XBOOLE_1:7;
      end;
      suppose
        j <> i; then
        x.j <> 1_Z & j in I by A3,FUNCT_7:32; then
        j in support(x,F) by A3,Def1;
        hence thesis by TARSKI:def 3,XBOOLE_1:7;
      end;
    end; then
    support(y,F) c= support(x,F) \/ {i};
    hence thesis by A1,A2,Th8;
  end;
