
theorem LM204L:
  for G, F being finite commutative Group, a be Element of G,
  f be Homomorphism of G, F
  holds the carrier of gr{f.a} = f.: (the carrier of gr{a})
  proof
    let G, F be finite commutative Group, a be Element of G,
    f be Homomorphism of G, F;
    for y be object
    holds y in the carrier of gr{f.a} iff y in f.: (the carrier of gr{a})
    proof
      let y be object;
      hereby
        assume
        AA1: y in the carrier of gr{f.a}; then
        reconsider y0 =y as Element of F by TARSKI:def 3, GROUP_2:def 5;
        y0 in gr{f.a} by AA1;
        then consider i be Element of NAT such that
        AA2: y0 =(f.a) |^i by GRCY26;
        AA3: y0 = f.(a |^i) by AA2, GROUP_6:37;
        a |^i in gr{a} by GRCY26;
        hence y in f.: (the carrier of gr{a}) by AA3, FUNCT_2:35;
      end;
      assume y in f.: (the carrier of gr{a});
      then consider x be object such that
      AA2: x in dom f & x in (the carrier of gr{a}) & y = f.x by FUNCT_1:def 6;
      reconsider x0 = x as Element of G by AA2;
      x0 in gr{a} by AA2;
      then consider i be Element of NAT such that
      AA3: x0 = a |^i by GRCY26;
      f.x0 = (f.a) |^i by AA3, GROUP_6:37;
      then f.x0 in gr{f.a} by GRCY26;
      hence y in the carrier of gr{f.a} by AA2;
    end;
    hence thesis by TARSKI:2;
  end;
