reserve x,y for set,
  G for Group,
  A,B,H,H1,H2 for Subgroup of G,
  a,b,c for Element of G,
  F,F1 for FinSequence of the carrier of G,
  I,I1 for FinSequence of INT,
  i,j for Element of NAT;

theorem
  for G being Group st ex F being FinSequence of the_normal_subgroups_of G st
  len F > 0 & F.1 = (Omega).G & F.(len F) = (1).G & for i st i in dom F
  & i+1 in dom F for G1 being strict normal Subgroup of G st G1 = F.i
  holds [.G1, (Omega).G.] = F.(i+1) holds G is nilpotent
proof
  let G be Group;
  given F being FinSequence of the_normal_subgroups_of G such that
A1: len F > 0 & F.1 = (Omega).G & F.(len F) = (1).G and
A2: for i st i in dom F & i+1 in dom F for G1 being strict normal
  Subgroup of G st G1 = F.i holds [.G1, (Omega).G.] = F.(i+1);
  for i st i in dom F & i+1 in dom F for H1,H2 being
  strict normal Subgroup of G st H1 = F.i & H2 = F.(i+1) holds H2 is
  Subgroup of H1 & [.H1, (Omega).G.] is Subgroup of H2
  proof
    let i;
    assume
A3: i in dom F & i+1 in dom F;
    let H1,H2 be strict normal Subgroup of G;
    assume H1 = F.i & H2 = F.(i+1);
    then H2 = [.H1, (Omega).G.] by A2,A3;
    hence thesis by Th15,GROUP_2:54;
  end;
  hence thesis by A1,Th21;
end;
