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;

theorem Th112:
  for X being Subset of REAL, r st X <> {} & for r9 st r9 in X
  holds r <= r9 holds lower_bound X >= r
proof
  let X be Subset of REAL, r such that
A1: X <> {} and
A2: for r9 st r9 in X holds r <= r9;
  for r9 be ExtReal st r9 in X holds r <= r9 by A2;
  then r is LowerBound of X by XXREAL_2:def 2;
  then
A3: X is bounded_below;
  now
    let r9 be Real;
    assume r9 > 0;
    then consider r1 be Real such that
A4: r1 in X and
A5: r1 < lower_bound X + r9 by A1,A3,Def2;
    r <= r1 by A2,A4;
    hence lower_bound X + r9 >= r by A5,XXREAL_0:2;
  end;
  hence thesis by XREAL_1:41;
end;
