consider r being Real such that
A1: for y being object st y in dom f holds
f . y < r by SEQ_2:def 1;
f | X is bounded_above

proof

take r ; :: according to SEQ_2:def 1 :: thesis: for b₁ being object holds
( not b₁ in dom (f | X) or not r <= (f | X) . b₁ )
let y be object ; :: thesis: ( not y in dom (f | X) or not r <= (f | X) . y )
assume A2: y in dom (f | X) ; :: thesis: not r <= (f | X) . y
then y in dom f by RELAT_1:57;
then f . y < r by A1;
hence not r <= (f | X) . y by A2, FUNCT_1:47; :: thesis: verum

end;

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