reserve            x for object,
               X,Y,Z for set,
         i,j,k,l,m,n for Nat,
                 r,s for Real,
                  no for Element of OrderedNAT,
                   A for Subset of [:NAT,NAT:];
reserve X,Y,X1,X2 for non empty set,
          cA1,cB1 for filter_base of X1,
          cA2,cB2 for filter_base of X2,
              cF1 for Filter of X1,
              cF2 for Filter of X2,
             cBa1 for basis of cF1,
             cBa2 for basis of cF2;
reserve T for non empty TopSpace,
        s for Function of [:NAT,NAT:], the carrier of T,
        M for Subset of the carrier of T;
reserve cF3,cF4 for Filter of the carrier of T;

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;
