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
  for X being non empty real-membered set, Y being real-membered set st
  X c= Y & Y is bounded_below holds lower_bound Y <= lower_bound X
proof
  let X be non empty real-membered set, Y be real-membered set;
  assume X c= Y & Y is bounded_below;
  then t in X implies t >= lower_bound Y by Def2;
  hence thesis by Th43;
end;
