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 Th125:
  for X,Y being Subset of REAL st Y is bounded_below & X <> {} &
  for r st r in X ex r1 st r1 in Y & r1 <= r holds
    lower_bound X >= lower_bound Y
proof
  let X,Y be Subset of REAL such that
A1: Y is bounded_below and
A2: X <> {} and
A3: for r st r in X ex r1 st r1 in Y & r1 <= r;
  now
    let r1;
    assume r1 in X;
    then consider r2 such that
A4: r2 in Y and
A5: r2 <= r1 by A3;
    lower_bound Y <= r2 by A1,A4,Def2;
    hence r1 >= lower_bound Y by A5,XXREAL_0:2;
  end;
  hence thesis by A2,Th112;
end;
