 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 Th44:
  for G1, G2, G3 being Group
  for f1 being Homomorphism of G1, G2
  for f2 being Homomorphism of G2, G3
  for g being Element of G1
  holds g in Ker (f2 * f1) iff f1.g in Ker(f2)
proof
  let G1, G2, G3 be Group;
  let f1 be Homomorphism of G1, G2;
  let f2 be Homomorphism of G2, G3;
  let g be Element of G1;
  thus g in Ker (f2 * f1) implies f1.g in Ker(f2)
  proof
    assume g in Ker (f2 * f1);
    then 1_(G3) = (f2 * f1).g by GROUP_6:41
               .= f2.(f1.g) by FUNCT_2:15;
    hence f1.g in Ker(f2) by GROUP_6:41;
  end;
  thus f1.g in Ker(f2) implies g in Ker (f2 * f1)
  proof
    assume f1.g in Ker(f2);
    then 1_G3 = f2.(f1.g) by GROUP_6:41
             .= (f2 * f1).g by FUNCT_2:15;
    hence g in Ker (f2 * f1) by GROUP_6:41;
  end;
end;
