let C be non empty set ; :: thesis: for X being set
for f being PartFunc of C,REAL st f | X is bounded_above holds
(max+ f) | X is bounded_above
let X be set ; :: thesis: for f being PartFunc of C,REAL st f | X is bounded_above holds
(max+ f) | X is bounded_above
let f be PartFunc of C,REAL ; :: thesis: ( f | X is bounded_above implies (max+ f) | X is bounded_above )
assume f | X is bounded_above ; :: thesis: (max+ f) | X is bounded_above
then consider b being real number such that
A1: for c being set st c in X /\ (dom f) holds
f . c <= b by RFUNCT_1:87;
A2: dom (max+ f) = dom f by RFUNCT_3:def 10;
ex r being Real st
for c being set st c in X /\ (dom (max+ f)) holds
(max+ f) . c <= r

proof

now

per cases ( b < 0 or b >= 0 ) ;

suppose A3: b < 0 ; :: thesis: ex r being Real st
for c being set st c in X /\ (dom (max+ f)) holds
(max+ f) . c <= r

for c being set st c in X /\ (dom (max+ f)) holds
(max+ f) . c <= 0

proof

let c be set ; :: thesis: ( c in X /\ (dom (max+ f)) implies (max+ f) . c <= 0 )
assume c in X /\ (dom (max+ f)) ; :: thesis: (max+ f) . c <= 0
then A4: c in X /\ (dom f) by RFUNCT_3:def 10;
then f . c <= b by A1;
then max (f . c),0 = 0 by A3, XXREAL_0:def 10;
then A5: max+ (f . c) = 0 by RFUNCT_3:def 1;
c in dom f by A4, XBOOLE_0:def 4;
hence (max+ f) . c <= 0 by A2, A5, RFUNCT_3:def 10; :: thesis: verum

end;

then consider r being Real such that
A6: ( r = 0 & ( for c being set st c in X /\ (dom (max+ f)) holds
(max+ f) . c <= r ) ) ;
thus ex r being Real st
for c being set st c in X /\ (dom (max+ f)) holds
(max+ f) . c <= r by A6; :: thesis: verum

end;

suppose A7: b >= 0 ; :: thesis: ex r being Real st
for c being set st c in X /\ (dom (max+ f)) holds
(max+ f) . c <= r

for c being set st c in X /\ (dom (max+ f)) holds
(max+ f) . c <= b

proof

let c be set ; :: thesis: ( c in X /\ (dom (max+ f)) implies (max+ f) . c <= b )
assume c in X /\ (dom (max+ f)) ; :: thesis: (max+ f) . c <= b
then A8: c in X /\ (dom f) by RFUNCT_3:def 10;
then f . c <= b by A1;
then max (f . c),0 <= b by A7, XXREAL_0:28;
then A9: max+ (f . c) <= b by RFUNCT_3:def 1;
c in dom f by A8, XBOOLE_0:def 4;
hence (max+ f) . c <= b by A2, A9, RFUNCT_3:def 10; :: thesis: verum

end;

then consider r being real number such that
A10: ( r = b & ( for c being set st c in X /\ (dom (max+ f)) holds
(max+ f) . c <= r ) ) ;
r is Real by XREAL_0:def 1;
hence ex r being Real st
for c being set st c in X /\ (dom (max+ f)) holds
(max+ f) . c <= r by A10; :: thesis: verum

end;

end;

end;

then consider r being Real such that
A11: for c being set st c in X /\ (dom (max+ f)) holds
(max+ f) . c <= r ;
thus ex r being Real st
for c being set st c in X /\ (dom (max+ f)) holds
(max+ f) . c <= r by A11; :: thesis: verum

end;

hence (max+ f) | X is bounded_above by RFUNCT_1:87; :: thesis: verum