let a, b, c, d, r be Real; :: thesis: for f being PartFunc of REAL,REAL st a <= c & c <= d & d <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f holds

( r (#) f is_integrable_on ['c,d'] & (r (#) f) | ['c,d'] is bounded )

let f be PartFunc of REAL,REAL; :: thesis: ( a <= c & c <= d & d <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f implies ( r (#) f is_integrable_on ['c,d'] & (r (#) f) | ['c,d'] is bounded ) )

assume that

A1: a <= c and

A2: ( c <= d & d <= b ) and

A3: ( f is_integrable_on ['a,b'] & f | ['a,b'] is bounded ) and

A4: ['a,b'] c= dom f ; :: thesis: ( r (#) f is_integrable_on ['c,d'] & (r (#) f) | ['c,d'] is bounded )

A5: f | ['c,d'] is bounded by A1, A2, A3, A4, INTEGRA6:18;

A6: ['c,d'] c= dom f by A2, A1, Th2, A4;

f is_integrable_on ['c,d'] by A1, A2, A3, A4, INTEGRA6:18;

hence r (#) f is_integrable_on ['c,d'] by A5, A6, INTEGRA6:9; :: thesis: (r (#) f) | ['c,d'] is bounded

thus (r (#) f) | ['c,d'] is bounded by A5, RFUNCT_1:80; :: thesis: verum

( r (#) f is_integrable_on ['c,d'] & (r (#) f) | ['c,d'] is bounded )

let f be PartFunc of REAL,REAL; :: thesis: ( a <= c & c <= d & d <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f implies ( r (#) f is_integrable_on ['c,d'] & (r (#) f) | ['c,d'] is bounded ) )

assume that

A1: a <= c and

A2: ( c <= d & d <= b ) and

A3: ( f is_integrable_on ['a,b'] & f | ['a,b'] is bounded ) and

A4: ['a,b'] c= dom f ; :: thesis: ( r (#) f is_integrable_on ['c,d'] & (r (#) f) | ['c,d'] is bounded )

A5: f | ['c,d'] is bounded by A1, A2, A3, A4, INTEGRA6:18;

A6: ['c,d'] c= dom f by A2, A1, Th2, A4;

f is_integrable_on ['c,d'] by A1, A2, A3, A4, INTEGRA6:18;

hence r (#) f is_integrable_on ['c,d'] by A5, A6, INTEGRA6:9; :: thesis: (r (#) f) | ['c,d'] is bounded

thus (r (#) f) | ['c,d'] is bounded by A5, RFUNCT_1:80; :: thesis: verum