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,G2 being strict normal Subgroup of G
  st G1 = F.i & G2 = F.(i+1) holds G2 is Subgroup of G1 &
  G./.G2 is cyclic Group 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,G2 being strict normal Subgroup
  of G st G1 = F.i & G2 = F.(i+1) holds G2 is Subgroup of G1 &
  G./.G2 is cyclic Group;
A3: 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 strict
  Subgroup of H1 & H1./.(H1,H2)`*` is Subgroup of center (G./.H2)
  proof
    let i;
    assume
A4: i in dom F & i+1 in dom F;
    let H1,H2 be strict normal Subgroup of G;
    assume
A5: H1 = F.i & H2 = F.(i+1);
    then H2 is strict Subgroup of H1 by A2,A4; then
A6: H1./.(H1,H2)`*` is Subgroup of G./.H2 by GROUP_6:28;
    G./.H2 is commutative Group by A2,A4,A5;
    hence thesis by A2,A4,A5,A6,GROUP_5:82;
  end;
  take F;
  thus thesis by A1,A3;
end;
