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;
