reserve i for Element of NAT;

theorem
  for G being strict commutative Group holds G is solvable
proof
  let G be strict commutative Group;
  (Omega).G in Subgroups G & (1).G in Subgroups G by GROUP_3:def 1;
  then <*(Omega).G,(1).G*>is FinSequence of Subgroups G by FINSEQ_2:13;
  then consider F being FinSequence of Subgroups G such that
A1: F=<*(Omega).G,(1).G*>;
A2: F.1=(Omega).G by A1;
A3: len F=2 by A1,FINSEQ_1:44;
A4: F.2=(1).G by A1;
  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 commutative
  proof
    let i;
    assume that
A5: i in dom F and
A6: i+1 in dom F;
    now
      let G1,G2 be strict Subgroup of G;
      assume
A7:   G1=F.i & G2=F.(i+1);
      dom F={1,2} by A3,FINSEQ_1:2,def 3;
      then
A8:   i=1 or i=2 by A5,TARSKI:def 2;
A9:   i+1 in{1,2} by A3,A6,FINSEQ_1:2,def 3;
      for N being normal Subgroup of G1 st N=G2 holds G1./.N is commutative
        by A2,A4,A7,A9,A8,GROUP_6:71,77,TARSKI:def 2;
      hence
      G2 is strict normal Subgroup of G1 & for N being normal Subgroup of
      G1 st N=G2 holds G1./.N is commutative
      by A1,A7,A9,A8,TARSKI:def 2;
    end;
    hence thesis;
  end;
  hence thesis by A3,A2,A4;
end;
