reserve X for set;
reserve G for Group;
reserve H for Subgroup of G;
reserve h,x,y for object;
reserve f for Endomorphism of G;
reserve phi for Automorphism of G;

theorem Th15:
  for G1,G2 being Group
  for f being Homomorphism of G1,G2
  for H being Subgroup of G1
  holds Ker(f|H) is Subgroup of Ker(f)
proof
  let G1,G2 be Group;
  let f be Homomorphism of G1,G2;
  let H be Subgroup of G1;
  A1: Ker(f|H) is Subgroup of G1 by GROUP_2:56;
  for g being Element of G1 st g in Ker(f|H) holds g in Ker(f)
  proof
    let g be Element of G1;
    assume A2: g in Ker(f|H);
    then A3: g in H by GROUP_2:40;
    (f|H).g = f.g by A2,Th1,GROUP_2:40;
    then 1_G2 = f.g by A2,A3,GROUP_6:41;
    hence g in Ker(f) by GROUP_6:41;
  end;
  hence thesis by A1,GROUP_2:58;
end;
