 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 ThJoinNormUnionRes:
  for I being set
  for J being Subset of I
  for F being normal Subgroup-Family of I,G
  for A being Subset of G
  st A = Union (Carrier (F|J))
  ex N being strict normal Subgroup of G
  st N = gr A
proof
  let I be set;
  let J be Subset of I;
  let F be normal Subgroup-Family of I,G;
  let A be Subset of G;
  assume A1: A = Union (Carrier (F|J));
  per cases;
  suppose J is empty;
    then Carrier (F|J) = {} --> (bool the carrier of G);
    then Union Carrier (F|J) = {} by FUNCT_6:26;
    then A2: A = {} the carrier of G by A1, SUBSET_1:def 2;
    take N = (1).G;
    thus thesis by A2, GROUP_4:30;
  end;
  suppose A3: J is non empty;
    then ex x being object st x in J by XBOOLE_0:def 1;
    then reconsider I as non empty set;
    reconsider J as non empty Subset of I by A3;
    reconsider F as normal Subgroup-Family of I,G;
    A = Union (Carrier (F|J)) by A1;
    hence thesis by LmJoinNormUnionRes;
  end;
end;
