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;
reserve k,n for Nat,
  r,r9,r1,r2 for Real,
  c,c9,c1,c2,c3 for Element of COMPLEX;
reserve z,z1,z2 for FinSequence of COMPLEX;
reserve x,z,z1,z2,z3 for Element of COMPLEX n,
  A,B for Subset of COMPLEX n;
reserve
  v,v1,v2 for FinSequence of REAL,
  n,m,k for Nat,
  x for set;

theorem Th132:
  for R being finite Subset of REAL holds R <> {} implies R is
bounded_above & upper_bound(R) in R & R is bounded_below & lower_bound(R) in R
proof
  let R be finite Subset of REAL;
  assume
A1: R <> {};
  P[card R] by Lm9;
  hence thesis by A1;
end;
