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 Th118:
  ex s19,s29 st s19 is_finer_than s1 & s29 is_finer_than s2 & s19
  is_equivalent_with s29
proof
  per cases;
  suppose
A1: len s1>1 & len s2>1;
    set s29=the_schreier_series_of(s2,s1);
    set s19=the_schreier_series_of(s1,s2);
    take s19,s29;
    thus s19 is_finer_than s1 & s29 is_finer_than s2 by A1,Th116;
    thus thesis by A1,Th117;
  end;
  suppose
A2: len s1<=1 or len s2<=1;
    per cases;
    suppose
A3:   len s1<=len s2;
      set s29=s2;
      set s19=s2;
      take s19,s29;
      thus s19 is_finer_than s1 & s29 is_finer_than s2 by A2,A3,Th114;
      thus thesis by Th113;
    end;
    suppose
A4:   len s1>len s2;
      set s29=s1;
      set s19=s1;
      take s19,s29;
      thus s19 is_finer_than s1 & s29 is_finer_than s2 by A2,A4,Th114;
      thus thesis by Th113;
    end;
  end;
end;
