reserve i for Element of NAT;

theorem
  for G being Group st ex F being FinSequence of Subgroups 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 Subgroup of G st G1=F.i & G2=F.(i+1) holds G2 is strict
normal Subgroup of G1 & for N being normal Subgroup of G1 st N=G2 holds G1./.N
  is cyclic Group holds G is solvable
proof
  let G be Group;
  given F being FinSequence of Subgroups G such that
A1: 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 Subgroup of G st G1=F.i & G2=F.(i+1) holds
G2 is strict normal Subgroup of G1 & for N being normal Subgroup of G1 st N=G2
  holds G1./.N is cyclic Group );
  take F;
  thus thesis by A1;
end;
