 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 ThStrSubFamResIsStr:
  for I being set
  for J being Subset of I
  for F being componentwise_strict Subgroup-Family of I,G
  holds F|J is componentwise_strict Subgroup-Family of J,G
proof
  let I be set;
  let J be Subset of I;
  let F be componentwise_strict Subgroup-Family of I,G;
  for i being object st i in J
  holds (F|J).i is strict Subgroup of G
  proof
    let i be object;
    assume A1: i in J;
    then A2: F.i is strict Subgroup of G by Def19;
    dom (F|J) = J by PARTFUN1:def 2;
    hence (F|J).i is strict Subgroup of G by A1,A2,FUNCT_1:47;
  end;
  hence F|J is componentwise_strict Subgroup-Family of J,G by Def19;
end;
