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 A1: rng f is bounded_below ; :: thesis: f | X is bounded_below
consider a being real number such that
A2: for y being real number st y in rng f holds
a <= y by A1, SEQ_4:def 2;
for x1 being set st x1 in X /\ (dom f) holds
a <= f . x1

proof

let x1 be set ; :: thesis: ( x1 in X /\ (dom f) implies a <= f . x1 )
assume A3: x1 in X /\ (dom f) ; :: thesis: a <= f . x1
X /\ (dom f) = dom f by XBOOLE_1:28;
then f . x1 in rng f by A3, FUNCT_1:def 5;
hence a <= f . x1 by A2; :: thesis: verum

end;

hence f | X is bounded_below by RFUNCT_1:88; :: thesis: verum