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
  for A being finite Subset of [:NAT,NAT:] holds ex n st
    A c= square-downarrow n
  proof
    let A be finite Subset of [:NAT,NAT:];
    consider m,n such that
A1: A c= [:Segm m,Segm n:] by Th16;
    reconsider m,n as Element of NAT by ORDINAL1:def 12;
    reconsider mn = max(m,n) as Nat;
    A c= square-downarrow mn
    proof
      Segm m c= Segm mn & Segm n c= Segm mn by XXREAL_0:25,NAT_1:39;
      then [:Segm m,Segm n:] c= [:Segm mn,Segm mn:] by ZFMISC_1:96;
      then [:Segm m,Segm n:] c= square-downarrow mn by Th30;
      hence thesis by A1;
    end;
    hence thesis;
  end;
