
theorem Th15:
  for I,J be non empty set,
      G be Group,
      x be Function of I,G,
      y be Function of J,G,
      a be Function of I,J
  st a is onto & x = y * a
  holds support(y) = a .: support(x)
  proof
    let I,J be non empty set,
        G be Group,
        x be Function of I,G,
        y be Function of J,G,
        a be Function of I,J;
    assume that
    A1: a is onto and
    A2: x = y * a;
    A3: rng a = J by A1,FUNCT_2:def 3;
    now
      let j be object;
      assume
      A4: j in support(y); then
      consider i be object such that
      A5: i in dom a and
      A6: j = a.i by A3,FUNCT_1:def 3;
      x.i = y.j by A2,A5,A6,FUNCT_1:13; then
      x.i <> 1_G by A4,GROUP_19:def 2; then
      i in support(x) by A5,GROUP_19:def 2;
      hence j in a .: support(x) by A5,A6,FUNCT_1:def 6;
    end; then
    A7: support(y) c= a .: support(x);
    now
      let j be object;
      assume j in a .: support(x); then
      consider i be object such that
      A8:  i in dom a and
      A9: i in support(x) and
      A10: j = a.i by FUNCT_1:def 6;
      A11: j in J by A8,A10,FUNCT_2:5;
      x.i = y.j by A2,A8,A10,FUNCT_1:13; then
      y.j <> 1_G by A9,GROUP_19:def 2;
      hence j in support(y) by A11,GROUP_19:def 2;
    end; then
    a .: support(x) c= support(y);
    hence thesis by A7,XBOOLE_0:def 10;
  end;
