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;
