 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 G being Group
  for N being finite Subset of the_normal_subgroups_of G
  for n being Nat st n = card N
  holds canFS N is normal Subgroup-Family of Seg n, G
proof
  let G be Group;
  let N be finite Subset of the_normal_subgroups_of G;
  let n be Nat;
  assume A1: n = card N;
  len (canFS N) = n by A1, FINSEQ_1:93;
  then A2: dom (canFS N) = Seg n by FINSEQ_1:def 3;
  the_normal_subgroups_of G c= Subgroups G by GRNILP_1:17;
  then A3: N is finite Subset of Subgroups G by XBOOLE_1:1;
  for i being object st i in Seg n
  holds (canFS N).i is normal Subgroup of G
  proof
    let i be object;
    assume i in Seg n;
    then (canFS N).i in rng (canFS N) by A2, FUNCT_1:3;
    then (canFS N).i in N by FINSEQ_1:def 4, TARSKI:def 3;
    hence thesis by GRNILP_1:def 1;
  end;
  hence canFS N is normal Subgroup-Family of (Seg n), G
  by A1, A3, Def18,ThCanSubgrFam;
end;
