
theorem Th7:
  for I,J be non empty set,
      a be Function of I,J,
      F be Group-Family of J,
      x be Function
  holds a .: support(x*a, F*a) c= support(x,F)
  proof
    let I,J be non empty set,
        a be Function of I,J,
        F be Group-Family of J,
        x be Function;
    for j be object st j in a .: support(x*a, F*a) holds j in support(x,F)
    proof
      let j be object;
      assume j in a .: support(x*a, F*a); then
      consider i be object such that
      A1: i in dom a and
      A2: i in support(x*a, F*a) and
      A3: j = a.i by FUNCT_1:def 6;
      consider Z being Group such that
B1:   Z = (F*a).i & (x*a).i <> 1_Z & i in I by A2,GROUP_19:def 1;
      reconsider y = x*a as Function;
      reconsider G = F*a as Group-Family of I;
      reconsider i as Element of I by B1;
      j = a.i by A3; then
      reconsider j as Element of J;
      A5: 1_G.i = 1_F.j by A1,A3,FUNCT_1:13;
      x.j <> 1_F.j by A1,A3,B1,A5,FUNCT_1:13;
      hence thesis by GROUP_19:def 1;
    end;
    hence thesis;
  end;
