reserve a,b,r,x,y for Real,
  i,j,k,n for Nat,
  x1 for set;

theorem
  for X,Y be non empty set, f be PartFunc of X,REAL st
    f|X is bounded_below & Y c= X holds f|Y|Y is bounded_below
proof
  let X,Y be non empty set;
  let f be PartFunc of X,REAL;
  assume f|X is bounded_below;
  then consider a be Real such that
A1: for x being object st x in X /\ dom f holds f.x>=a by RFUNCT_1:71;
  assume
A2: Y c= X;
  for x being object st x in Y /\ dom (f|Y) holds (f|Y).x>=a
  proof
    let x be object;
A3: dom f /\ Y c= dom f /\ X by A2,XBOOLE_1:26;
    assume x in Y /\ dom (f|Y);
    then
A4: x in dom (f|Y) by XBOOLE_0:def 4;
    then x in dom f /\ Y by RELAT_1:61;
    then a <= f.x by A1,A3;
    hence thesis by A4,FUNCT_1:47;
  end;
  hence thesis by RFUNCT_1:71;
end;
