let X be non empty set ; :: thesis: for f being Function of X,REAL st f is with_min holds
- f is with_max
let f be Function of X,REAL; :: thesis: ( f is with_min implies - f is with_max )
assume that
A1: f .: X is bounded_below and
A2: lower_bound (f .: X) in f .: X ; :: according to MEASURE6:def 9,MEASURE6:def 17 :: thesis: - f is with_max
A3: -- (f .: X) = (- f) .: X by MEASURE6:101;
hence (- f) .: X is bounded_above by A1, Lm2; :: according to MEASURE6:def 8,MEASURE6:def 16 :: thesis: upper_bound ((- f) .: X) in (- f) .: X
then A4: upper_bound ((- f) .: X) = - (lower_bound (-- (-- (f .: X)))) by A3, MEASURE6:80
.= - (lower_bound (f .: X)) ;
- (lower_bound (f .: X)) in -- (f .: X) by A2, MEMBER_1:11;
hence upper_bound ((- f) .: X) in (- f) .: X by A4, MEASURE6:101; :: thesis: verum