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 Th31:
  for x being Element of [: base_of_frechet_filter,base_of_frechet_filter :]
  holds ex i,j st x = [:NAT \ Segm i,NAT \ Segm j:]
  proof
    let x be Element of [: base_of_frechet_filter,base_of_frechet_filter :];
    x in the set of all [:B1,B2:] where B1 is Element of
    base_of_frechet_filter, B2 is Element of base_of_frechet_filter;
    then consider B1 be Element of base_of_frechet_filter,
                  B2 be Element of base_of_frechet_filter such that
A1: x = [:B1,B2:];
    consider i such that
A2: B1 = NAT \ Segm i by Th20;
    consider j such that
A3: B2 = NAT \ Segm j by Th20;
    take i,j;
    thus thesis by A1,A2,A3;
  end;
