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 5,MEASURE6:def 13 :: thesis: - f is with_max
A3: -- (f .: X) = (- f) .: X by Th65;
hence (- f) .: X is bounded_above by A1, Lm3; :: according to MEASURE6:def 4,MEASURE6:def 12 :: thesis: upper_bound ((- f) .: X) in (- f) .: X
then A4: upper_bound ((- f) .: X) = - (lower_bound (-- (-- (f .: X)))) by A3, Th44
.= - (lower_bound (f .: X)) ;
- (lower_bound (f .: X)) in -- (f .: X) by A2;
hence upper_bound ((- f) .: X) in (- f) .: X by A4, Th65; :: thesis: verum