 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
  ex S being componentwise_strict normal Subgroup-Family of F
  st for i being Element of I holds S.i = (F.i)`
proof
  deffunc Fun(Group) = ($1)`;
  A1: for G being Group holds Fun(G) is strict Subgroup of G;
  consider S being componentwise_strict Subgroup-Family of F such that
  A2: for i being Element of I holds S.i = Fun(F.i)
  from StrSubFamSch(A1);
  for i being Element of I holds S.i is normal Subgroup of F.i
  proof
    let i be Element of I;
    S.i = (F.i)` by A2;
    hence S.i is normal Subgroup of F.i;
  end;
  then reconsider S as componentwise_strict normal Subgroup-Family of F
  by Def7;
  take S;
  thus thesis by A2;
end;
