consider r being Real such that
A1: for y being object st y in dom f holds
r < f . y by SEQ_2:def 2;
f | X is bounded_below

proof

take r ; :: according to SEQ_2:def 2 :: thesis: for b₁ being object holds
( not b₁ in dom (f | X) or not (f | X) . b₁ <= r )
let y be object ; :: thesis: ( not y in dom (f | X) or not (f | X) . y <= r )
assume A2: y in dom (f | X) ; :: thesis: not (f | X) . y <= r
then y in dom f by RELAT_1:57;
then r < f . y by A1;
hence not (f | X) . y <= r by A2, FUNCT_1:47; :: thesis: verum

end;

hence for b₁ being real-valued Function st b₁ = f | X holds
b₁ is bounded_below ; :: thesis: verum