reserve X for set;
reserve G for Group;
reserve H for Subgroup of G;
reserve h,x,y for object;
reserve f for Endomorphism of G;
reserve phi for Automorphism of G;
reserve K for characteristic Subgroup of G;

theorem
  for G being Group
  for X being finite set
  st X <> {} & (for A being Element of X
                ex N being strict normal Subgroup of G
                st A = the carrier of N)
  ex N being strict normal Subgroup of G
  st the carrier of N = meet X
proof
  let G be Group;
  let X be finite set;
  assume A1: X <> {};
  assume A2: for A being Element of X
             ex N being strict normal Subgroup of G
             st A = the carrier of N;
  defpred P[Group] means $1 is normal Subgroup of G & the carrier of $1 in X;
  set Fam = {A where A is Subset of G : ex N being strict Subgroup of G
                                        st A = the carrier of N & P[N]};
  set Fam2 = {A where A is Subset of G : ex N being strict Subgroup of G
                                         st A = the carrier of N &
                                         N is normal & P[N]};
  A3: ex H being strict Subgroup of G st P[H]
  proof
    consider A being object such that
    B1: A in X by A1,XBOOLE_0:def 1;
    reconsider A as Element of X by B1;
    consider H being strict normal Subgroup of G such that
    B2: A = the carrier of H by A2;
    take H;
    thus P[H] by B1,B2;
  end;
  consider N being strict Subgroup of G such that
  A4: the carrier of N = meet Fam from GROUP_4:sch 1(A3);
  for A being object holds A in Fam iff A in Fam2
  proof
    let A be object;
    thus A in Fam implies A in Fam2
    proof
      assume A in Fam;
      then consider A0 being Subset of G such that
      B1: A = A0 and
      B2: ex N being strict Subgroup of G st A0 = the carrier of N & P[N];
      consider N being strict Subgroup of G such that
      B3: A0 = the carrier of N & P[N] by B2;
      thus A in Fam2 by B1,B3;
    end;
    thus A in Fam2 implies A in Fam
    proof
      assume A in Fam2;
      then consider A0 being Subset of G such that
      B1: A = A0 &
          ex N being strict Subgroup of G
          st A0 = the carrier of N & N is normal & P[N];
      thus A in Fam by B1;
    end;
  end;
  then A5: Fam = Fam2 by TARSKI:2;

  A6: ex H being strict normal Subgroup of G st P[H] by A3;
  for H being strict Subgroup of G st the carrier of H = meet Fam2
  holds H is strict normal Subgroup of G
  from MeetOfNormsIsNormal(A6);
  then reconsider N as strict normal Subgroup of G by A4,A5;
  take N;

  for A being object holds A in Fam iff A in X
  proof
    let A be object;
    thus A in Fam implies A in X
    proof
      assume A in Fam;
      then consider A0 being Subset of G such that
      B1: A = A0 &
          ex N being strict Subgroup of G st A0 = the carrier of N & P[N];
      thus thesis by B1;
    end;
    thus A in X implies A in Fam
    proof
      assume B1: A in X;
      then consider N being strict normal Subgroup of G such that
      B2: A = the carrier of N by A2;
      A is Subset of G by B2,GROUP_2:def 5;
      hence A in Fam by B1,B2;
    end;
  end;
  hence the carrier of N = meet X by A4,TARSKI:2;
end;
