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

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

thus ( f | Y is bounded_below implies ex r being Real st
for c being object st c in Y /\ (dom f) holds
r <= f . c ) :: thesis: ( ex r being Real st
for c being object st c in Y /\ (dom f) holds
r <= f . c implies f | Y is bounded_below )
proof
given r being Real such that A1: for p being object st p in dom (f | Y) holds
r < (f | Y) . p ; :: according to SEQ_2:def 2 :: thesis: ex r being Real st
for c being object st c in Y /\ (dom f) holds
r <= f . c

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

let c be object ; :: thesis: ( c in Y /\ (dom f) implies r <= f . c )
assume c in Y /\ (dom f) ; :: thesis: r <= f . c
then A2: c in dom (f | Y) by RELAT_1:61;
then r < (f | Y) . c by A1;
hence r <= f . c by ; :: thesis: verum
end;
given r being Real such that A3: for c being object st c in Y /\ (dom f) holds
r <= f . c ; :: thesis: f | Y is bounded_below
reconsider r1 = r - 1 as Real ;
take r1 ; :: according to SEQ_2:def 2 :: thesis: for b1 being object holds
( not b1 in dom (f | Y) or not (f | Y) . b1 <= r1 )

let p be object ; :: thesis: ( not p in dom (f | Y) or not (f | Y) . p <= r1 )
assume A4: p in dom (f | Y) ; :: thesis: not (f | Y) . p <= r1
then p in Y /\ (dom f) by RELAT_1:61;
then r <= f . p by A3;
then A5: r <= (f | Y) . p by ;
r1 < r by XREAL_1:44;
hence not (f | Y) . p <= r1 by ; :: thesis: verum