let n be Element of NAT ; :: thesis: for a, b, c, d being Real
for f, g being PartFunc of REAL,() st a <= 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 & c in ['a,b'] & d in ['a,b'] holds
integral ((f - g),c,d) = (integral (f,c,d)) - (integral (g,c,d))

let a, b, c, d be Real; :: thesis: for f, g being PartFunc of REAL,() st a <= 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 & c in ['a,b'] & d in ['a,b'] holds
integral ((f - g),c,d) = (integral (f,c,d)) - (integral (g,c,d))

let f, g be PartFunc of REAL,(); :: thesis: ( a <= 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 & c in ['a,b'] & d in ['a,b'] implies integral ((f - g),c,d) = (integral (f,c,d)) - (integral (g,c,d)) )
assume A1: ( a <= 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 & c in ['a,b'] & d in ['a,b'] ) ; :: thesis: integral ((f - g),c,d) = (integral (f,c,d)) - (integral (g,c,d))
reconsider f1 = f, g1 = g as PartFunc of REAL,(REAL n) by REAL_NS1:def 4;
A2: f1 | ['a,b'] is bounded by ;
A3: g1 | ['a,b'] is bounded by ;
A4: f1 is_integrable_on ['a,b'] by Th43, A2, A1;
A5: g1 is_integrable_on ['a,b'] by Th43, A3, A1;
A6: f1 - g1 = f - g by NFCONT_4:10;
A7: ( integral (f1,c,d) = integral (f,c,d) & integral (g1,c,d) = integral (g,c,d) ) by ;
A8: ['a,b'] c= dom (f - g) by A1, A6, Th5;
( f1 - g1 is_integrable_on ['a,b'] & (f1 - g1) | ['a,b'] is bounded ) by A1, A2, A3, A4, A5, Th13;
then ( f - g is_integrable_on ['a,b'] & (f - g) | ['a,b'] is bounded ) by Th43, A6, A8, Th34;
hence integral ((f - g),c,d) = integral ((f1 - g1),c,d) by
.= (integral (f1,c,d)) - (integral (g1,c,d)) by A1, A2, A3, A4, A5, Th28
.= (integral (f,c,d)) - (integral (g,c,d)) by ;
:: thesis: verum