let Y be set ; :: thesis: for f being real-valued Function holds
( f | Y is bounded_above iff ex r being real number st
for c being set st c in Y /\ (dom f) holds
f . c <= r )
let f be real-valued Function; :: thesis: ( f | Y is bounded_above iff ex r being real number st
for c being set st c in Y /\ (dom f) holds
f . c <= r )
thus ( f | Y is bounded_above implies ex r being real number st
for c being set st c in Y /\ (dom f) holds
f . c <= r ) :: thesis: ( ex r being real number st
for c being set st c in Y /\ (dom f) holds
f . c <= r implies f | Y is bounded_above )

proof

given r being real number such that G: for p being set st p in dom (f | Y) holds
(f | Y) . p < r ; :: according to SEQ_2:def 1 :: thesis: ex r being real number st
for c being set st c in Y /\ (dom f) holds
f . c <= r
take r ; :: thesis: for c being set st c in Y /\ (dom f) holds
f . c <= r
let c be set ; :: thesis: ( c in Y /\ (dom f) implies f . c <= r )
assume c in Y /\ (dom f) ; :: thesis: f . c <= r
then Z: c in dom (f | Y) by RELAT_1:90;
then (f | Y) . c < r by G;
hence f . c <= r by Z, FUNCT_1:70; :: thesis: verum

end;

given r being real number such that G: for c being set st c in Y /\ (dom f) holds
f . c <= r ; :: thesis: f | Y is bounded_above
reconsider r1 = r + 1 as real number ;
A: r < r1 by XREAL_1:31;
take r1 ; :: according to SEQ_2:def 1 :: thesis: for b₁ being set holds
( not b₁ in dom (f | Y) or not r1 <= (f | Y) . b₁ )
let p be set ; :: thesis: ( not p in dom (f | Y) or not r1 <= (f | Y) . p )
assume Z: p in dom (f | Y) ; :: thesis: not r1 <= (f | Y) . p
then p in Y /\ (dom f) by RELAT_1:90;
then f . p <= r by G;
then (f | Y) . p <= r by Z, FUNCT_1:70;
hence not r1 <= (f | Y) . p by A, XXREAL_0:2; :: thesis: verum