consider r being real number such that
W: for y being set 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 set holds
( not b₁ in dom (f | X) or not r <= (f | X) . b₁ )
let y be set ; :: thesis: ( not y in dom (f | X) or not r <= (f | X) . y )
assume Z: y in dom (f | X) ; :: thesis: not r <= (f | X) . y
then y in dom f by RELAT_1:86;
then f . y < r by W;
hence not r <= (f | X) . y by Z, FUNCT_1:70; :: thesis: verum

end;

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