let f be real-valued Function; :: thesis: ( rng f is bounded_below implies f is bounded_below )
set X = dom f;
assume rng f is bounded_below ; :: thesis: f is bounded_below
then consider a being Real such that
A1: a is LowerBound of rng f ;
AA: f | (dom f) = f ;
for x1 being object st x1 in (dom f) /\ (dom f) holds
a <= f . x1

proof

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

end;

hence f is bounded_below by AA, RFUNCT_1:71; :: thesis: verum