 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 I being set
  for F being normal Subgroup-Family of I,G
  for A being Subset of G
  st A = Union (Carrier F)
  ex N being strict normal Subgroup of G
  st N = gr A
proof
  let I be set;
  let F be normal Subgroup-Family of I,G;
  let A be Subset of G;
  assume A1: A = Union (Carrier F);
  reconsider J=[#] I as Subset of I;
  I = J by SUBSET_1:def 3;
  then A = Union (Carrier (F|J)) by A1;
  hence thesis by ThJoinNormUnionRes;
end;
