 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 J being non empty Subset of I
  for F being normal Subgroup-Family of I,G
  holds F|J is normal Subgroup-Family of J,G
proof
  let J be non empty Subset of I;
  let F be normal Subgroup-Family of I,G;
  for j being Element of J holds (F|J).j is normal Subgroup of G
  proof
    let j be Element of J;
    j in J;
    then j in dom (F|J) by PARTFUN1:def 2;
    then F.j = (F|J).j by FUNCT_1:47;
    hence (F|J).j is normal Subgroup of G;
  end;

  hence F|J is normal Subgroup-Family of J,G by ThS1;
end;
