let a, b, c, d, r be Real; :: thesis: 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; :: thesis: 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; :: thesis: ( 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 ) ; :: thesis: ( 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; :: thesis: (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; :: thesis: verum

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; :: thesis: 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; :: thesis: ( 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 ) ; :: thesis: ( 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; :: thesis: (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; :: thesis: verum