reserve x,O for set,
  o for Element of O,
  G,H,I for GroupWithOperators of O,
  A, B for Subset of G,
  N for normal StableSubgroup of G,
  H1,H2,H3 for StableSubgroup of G,
  g1,g2 for Element of G,
  h1,h2 for Element of H1,
  h for Homomorphism of G,H;
reserve E for set,
  A for Action of O,E,
  C for Subset of G,
  N1 for normal StableSubgroup of H1;
reserve y for set,
  H19,H29 for StableSubgroup of G,
  N19 for normal StableSubgroup of H19,
  s1,s19,s2,s29 for CompositionSeries of G,
  fs for FinSequence of the_stable_subgroups_of G,
  f1,f2 for FinSequence,
  i,j,n for Nat;

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;
