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;
