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

thus ( f | Y is bounded_above implies ex r being Real st
for c being object st c in Y /\ (dom f) holds
f . c <= r ) :: thesis: ( ex r being Real st
for c being object st c in Y /\ (dom f) holds
f . c <= r implies f | Y is bounded_above )
proof
given r being Real such that A1: for p being object st p in dom (f | Y) holds
(f | Y) . p < r ; :: according to SEQ_2:def 1 :: thesis: ex r being Real st
for c being object st c in Y /\ (dom f) holds
f . c <= r

take r ; :: thesis: for c being object st c in Y /\ (dom f) holds
f . c <= r

let c be object ; :: thesis: ( c in Y /\ (dom f) implies f . c <= r )
assume c in Y /\ (dom f) ; :: thesis: f . c <= r
then A2: c in dom (f | Y) by RELAT_1:61;
then (f | Y) . c < r by A1;
hence f . c <= r by ; :: thesis: verum
end;
given r being Real such that A3: for c being object st c in Y /\ (dom f) holds
f . c <= r ; :: thesis: f | Y is bounded_above
reconsider r1 = r + 1 as Real ;
take r1 ; :: according to SEQ_2:def 1 :: thesis: for b1 being object holds
( not b1 in dom (f | Y) or not r1 <= (f | Y) . b1 )

let p be object ; :: thesis: ( not p in dom (f | Y) or not r1 <= (f | Y) . p )
assume A4: p in dom (f | Y) ; :: thesis: not r1 <= (f | Y) . p
then p in Y /\ (dom f) by RELAT_1:61;
then f . p <= r by A3;
then A5: (f | Y) . p <= r by ;
r < r1 by XREAL_1:29;
hence not r1 <= (f | Y) . p by ; :: thesis: verum