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;

theorem Th24:
  ex cB1 st <.cB1.) = cF1
  proof
A1: cF1 is basis of cF1 by CARDFIL2:13;
    then reconsider cB1 = cF1 as filter_base of X1 by CARDFIL2:29;
    <.cB1.) = <.cB1.];
    hence thesis by A1,Th23;
  end;
