let X be non empty set ; :: thesis: for f being PartFunc of X,REAL st rng f is bounded_below holds
f | X is bounded_below
let f be PartFunc of X,REAL; :: thesis: ( rng f is bounded_below implies f | X is bounded_below )
assume rng f is bounded_below ; :: thesis: f | X is bounded_below
then consider a being Real such that
A1: a is LowerBound of rng f ;
for x1 being object st x1 in X /\ (dom f) holds
a <= f . x1

proof

let x1 be object ; :: thesis: ( x1 in X /\ (dom f) implies a <= f . x1 )
A2: X /\ (dom f) = dom f by XBOOLE_1:28;
assume x1 in X /\ (dom f) ; :: thesis: a <= f . x1
then f . x1 in rng f by A2, FUNCT_1:def 3;
hence a <= f . x1 by A1, XXREAL_2:def 2; :: thesis: verum

end;

hence f | X is bounded_below by RFUNCT_1:71; :: thesis: verum