theorem
  f1=the_series_of_quotients_of s1 & i in dom f1 & (for H st H = f1.i
holds H is trivial) implies Del(s1,i) is CompositionSeries of G & for s2 st s2
  = Del(s1,i) holds the_series_of_quotients_of s2 = Del(f1,i)
proof
  assume
A1: f1=the_series_of_quotients_of s1;
  assume
A2: i in dom f1;
  assume
A3: for H st H = f1.i holds H is trivial;
  then
A4: s1.i=s1.(i+1) by A1,A2,Th103;
A5: i in dom s1 & i+1 in dom s1 by A1,A2,A3,Th103;
  hence Del(s1,i) is CompositionSeries of G by A4,Th94,FINSEQ_3:105;
  let s2;
  assume s2 = Del(s1,i);
  hence thesis by A1,A5,A4,Th104;
end;
