let a, b, c, d, r be Real; for Y being RealBanachSpace
for f being continuous PartFunc of REAL, the carrier of Y st a <= c & c <= d & d <= b & ['a,b'] c= dom f holds
( r (#) f is_integrable_on ['c,d'] & (r (#) f) | ['c,d'] is bounded )
let Y be RealBanachSpace; for f being continuous PartFunc of REAL, the carrier of Y st a <= c & c <= d & d <= b & ['a,b'] c= dom f holds
( r (#) f is_integrable_on ['c,d'] & (r (#) f) | ['c,d'] is bounded )
let f be continuous PartFunc of REAL, the carrier of Y; ( a <= c & c <= d & d <= b & ['a,b'] c= dom f implies ( r (#) f is_integrable_on ['c,d'] & (r (#) f) | ['c,d'] is bounded ) )
assume A1:
( a <= c & c <= d & d <= b & ['a,b'] c= dom f )
; ( r (#) f is_integrable_on ['c,d'] & (r (#) f) | ['c,d'] is bounded )
reconsider A = ['c,d'] as non empty closed_interval Subset of REAL ;
A2:
f is_integrable_on A
by A1, Th1909;
A c= dom f
by A1, INTEGR19:2;
hence
r (#) f is_integrable_on ['c,d']
by A2, INTEGR18:13; (r (#) f) | ['c,d'] is bounded
a <= d
by A1, XXREAL_0:2;
then
f | ['a,b'] is bounded
by A1, Th1, XXREAL_0:2;
then
f | ['c,d'] is bounded
by A1, Th1915b;
hence
(r (#) f) | ['c,d'] is bounded
by Th1935a; verum