let n be Element of NAT ; :: thesis: for a, b, c, d being Real
for f, g being PartFunc of REAL,(REAL n) 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 a, b, c, d be Real; :: thesis: for f, g being PartFunc of REAL,(REAL n) 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 n); :: 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 that
A1: ( a <= c & c <= d & d <= b ) and
A2: ( f is_integrable_on ['a,b'] & g is_integrable_on ['a,b'] & f | ['a,b'] is bounded & g | ['a,b'] is bounded ) and
A3: ( ['a,b'] c= dom f & ['a,b'] c= dom g ) ; :: thesis: ( f - g is_integrable_on ['c,d'] & (f - g) | ['c,d'] is bounded )
A4: dom g = dom (- g) by NFCONT_4:def 3;
A5: f - g = f + (- g) by Lm1;
a <= d by A1, XXREAL_0:2;
then a <= b by A1, XXREAL_0:2;
then ( - g is_integrable_on ['a,b'] & (- g) | ['a,b'] is bounded ) by A2, A3, Th12;
hence ( f - g is_integrable_on ['c,d'] & (f - g) | ['c,d'] is bounded ) by A5, A1, A2, A3, A4, Th10; :: thesis: verum