 reserve I for non empty set;
 reserve i for Element of I;
 reserve F for Group-Family of I;
 reserve G for Group;
reserve S for Subgroup-Family of F;
reserve f for Homomorphism-Family of G, F;

theorem
  for G1, G2, G3 being Group
  for f1 being Homomorphism of G1, G2
  for f2 being Homomorphism of G2, G3
  holds the carrier of Ker (f2 * f1) = f1" (the carrier of Ker f2)
proof
  let G1, G2, G3 be Group;
  let f1 be Homomorphism of G1, G2;
  let f2 be Homomorphism of G2, G3;
  A1: f1 "(the carrier of Ker f2) is Subset of G1
      & the carrier of Ker (f2 * f1) is Subset of G1
  by GROUP_2:def 5, FUNCT_2:39;
  for g being Element of G1
  st g in the carrier of Ker (f2 * f1)
  holds g in f1 " (the carrier of (Ker f2))
  proof
    let g be Element of G1;
    assume g in the carrier of Ker (f2 * f1);
    then g in Ker (f2 * f1);
    then f1.g in Ker f2 by Th44;
    hence g in f1 " (the carrier of (Ker f2)) by FUNCT_2:38;
  end; then
  A2: the carrier of Ker (f2 * f1) c= f1" (the carrier of Ker f2)
  by A1, SUBSET_1:2;
  for g being Element of G1
  st g in f1 " (the carrier of (Ker f2))
  holds g in the carrier of Ker (f2 * f1)
  proof
    let g be Element of G1;
    assume g in f1 " (the carrier of (Ker f2));
    then f1.g in Ker f2 by FUNCT_2:38;
    then g in Ker (f2 * f1) by Th44;
    hence g in the carrier of Ker (f2 * f1);
  end; then
  A3: f1" (the carrier of Ker f2) c= the carrier of Ker (f2 * f1)
  by A1, SUBSET_1:2;
  thus the carrier of Ker (f2 * f1) = f1" (the carrier of Ker f2)
  by A2, A3, XBOOLE_0:def 10;
end;
