 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;

theorem
  for S being Group-Family of I
  holds S is Subgroup-Family of F
  iff (for i being Element of I holds S.i is Subgroup of F.i)
proof
  let S be Group-Family of I;
  thus (S is Subgroup-Family of F) implies (for i being Element of I
  holds S.i is Subgroup of F.i)
  proof
    assume A1: S is Subgroup-Family of F;
    let i be Element of I;
    reconsider SS=S as Subgroup-Family of F by A1;
    S.i = SS.i;
    hence thesis;
  end;
  assume for i being Element of I holds S.i is Subgroup of F.i;
  then S is F-Subgroup-yielding;
  hence S is Subgroup-Family of F;
end;
