theorem Th42:
  s"(M) in <. Frechet_Filter(NAT),Frechet_Filter(NAT).) iff
  ex n st square-uparrow n c= s"(M)
  proof
    hereby
      assume s"(M) in <. Frechet_Filter(NAT),Frechet_Filter(NAT).);
      then consider b be Element of
      [: base_of_frechet_filter,base_of_frechet_filter:] such that
A1:   b c= s"(M) by Th35,CARDFIL2:def 8;
      ex n st square-uparrow n c= b by Th32;
      hence ex n st square-uparrow n c= s"(M) by A1,XBOOLE_1:1;
    end;
    given n such that
A2: square-uparrow n c= s"(M);
    square-uparrow n in the set of all square-uparrow n where n is Nat;
    then ex b2 be Element of
      [: base_of_frechet_filter,base_of_frechet_filter:] st
      b2 c= square-uparrow n by Th34;
    then
A3: ex b2 be Element of [: base_of_frechet_filter,base_of_frechet_filter:] st
      b2 c= s"(M) by A2,XBOOLE_1:1;
    dom s = [:NAT,NAT:] by FUNCT_2:def 1;
    then s"(M) is Subset of [:NAT,NAT:] by RELAT_1:132;
    hence s"(M) in <. Frechet_Filter(NAT),Frechet_Filter(NAT).)
      by A3,Th35,CARDFIL2:def 8;
  end;
