reserve n,k,k1,m,m1,n1,n2,l for Nat;
reserve r,r1,r2,p,p1,g,g1,g2,s,s1,s2,t for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve Nseq for increasing sequence of NAT;
reserve x for set;
reserve X,Y for Subset of REAL;

theorem
  X is real-bounded non empty implies lower_bound X <= upper_bound X
proof
  assume X is real-bounded non empty;
  then reconsider X as real-bounded non empty real-membered set;
  lower_bound X <= upper_bound X by XXREAL_2:40;
  hence thesis;
end;
