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 )

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 b_{1} being object holds

( not b_{1} in dom (f | Y) or not (f | Y) . b_{1} <= 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 A4, FUNCT_1:47;

r1 < r by XREAL_1:44;

hence not (f | Y) . p <= r1 by A5, XXREAL_0:2; :: thesis: verum

( 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 A3:
for c being object st c in Y /\ (dom f) holds
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 A2, FUNCT_1:47; :: thesis: verum

end;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 A2, FUNCT_1:47; :: thesis: verum

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 b

( not b

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 A4, FUNCT_1:47;

r1 < r by XREAL_1:44;

hence not (f | Y) . p <= r1 by A5, XXREAL_0:2; :: thesis: verum