 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 ThSubFamRes:
  for I being set
  for F being Subgroup-Family of I,G
  for J being set st J c= I
  holds F|J is Subgroup-Family of J,G
proof
  let I be set;
  let F be Subgroup-Family of I,G;
  let J be set;
  assume A1: J c= I;
  for j being object st j in J holds (F|J).j is Subgroup of G
  proof
    let j be object;
    assume A2: j in J;
    then F.j is Subgroup of G by A1, GROUP_20:def 1;
    hence (F|J).j is Subgroup of G by A2, FUNCT_1:49;
  end;
  hence F|J is Subgroup-Family of J,G by A1,GROUP_20:def 1;
end;
