let a, b, c, d be Real; :: thesis: for f, g being PartFunc of REAL,REAL st a <= c & c <= d & d <= b & f is_integrable_on ['a,b'] & g is_integrable_on ['a,b'] & f | ['a,b'] is bounded & g | ['a,b'] is bounded & ['a,b'] c= dom f & ['a,b'] c= dom g holds
( f - g is_integrable_on ['c,d'] & (f - g) | ['c,d'] is bounded )
let f, g be PartFunc of REAL,REAL; :: thesis: ( a <= c & c <= d & d <= b & f is_integrable_on ['a,b'] & g is_integrable_on ['a,b'] & f | ['a,b'] is bounded & g | ['a,b'] is bounded & ['a,b'] c= dom f & ['a,b'] c= dom g implies ( f - g is_integrable_on ['c,d'] & (f - g) | ['c,d'] is bounded ) )
assume A1: ( a <= c & c <= d & d <= b & f is_integrable_on ['a,b'] & g is_integrable_on ['a,b'] & f | ['a,b'] is bounded & g | ['a,b'] is bounded & ['a,b'] c= dom f & ['a,b'] c= dom g ) ; :: thesis: ( f - g is_integrable_on ['c,d'] & (f - g) | ['c,d'] is bounded )
A2: ((- 1) (#) g) | ['a,b'] is bounded by A1, RFUNCT_1:80;
A3: ['a,b'] c= dom ((- 1) (#) g) by A1, VALUED_1:def 5;
(- 1) (#) g is_integrable_on ['a,b'] by A1, INTEGRA6:9;
hence ( f - g is_integrable_on ['c,d'] & (f - g) | ['c,d'] is bounded ) by A1, A2, A3, INTEGRA6:19; :: thesis: verum