
theorem Th8:
  for I,J be non empty set,
      a be Function of I,J,
      F be Group-Family of J,
      x be Function
  st a is onto
  holds support(x,F) c= a .: support(x*a,F*a)
  proof
    let I,J be non empty set,
        a be Function of I,J,
        F be Group-Family of J,
        x be Function;
    assume
    A1: a is onto;
    for j be object st j in support(x,F) holds j in a .: support(x*a,F*a)
    proof
      let j be object;
      assume
      A2: j in support(x,F);
      A3: dom a = I by FUNCT_2:def 1;
      rng a = J by A1,FUNCT_2:def 3; then
      consider i be object such that
      A4: i in I and
      A5: j = a.i by A2,FUNCT_2:11;
      reconsider y = x*a as Function;
      reconsider G = F*a as Group-Family of I;
      consider Z being Group such that
B1:   Z = F.j & x.j <> 1_Z & j in J by A2,GROUP_19:def 1;
      reconsider j as Element of J by B1;
      reconsider i as Element of I by A4;
      A6: 1_G.i = 1_F.j by A3,A5,FUNCT_1:13;
      x.j = y.i by A3,A5,FUNCT_1:13; then
      i in support(y,G) by A6,B1,GROUP_19:def 1;
      hence thesis by A3,A5,FUNCT_1:def 6;
    end;
    hence thesis;
  end;
