let n be Element of NAT ; for a, b, c, d being Real
for f, g being PartFunc of REAL,(REAL-NS n) 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; for f, g being PartFunc of REAL,(REAL-NS n) 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,(REAL-NS n); ( 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'] )
; 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 Th34, A1;
A3:
g1 | ['a,b'] is bounded
by Th34, A1;
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_4:5;
A7:
( integral (f1,c,d) = integral (f,c,d) & integral (g1,c,d) = integral (g,c,d) )
by A1, Th48;
A8:
['a,b'] c= dom (f + g)
by A1, A6, Th4;
( f1 + g1 is_integrable_on ['a,b'] & (f1 + g1) | ['a,b'] is bounded )
by A1, A2, A3, A4, A5, Th10;
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 A1, A8, Th48, NFCONT_4:5
.=
(integral (f1,c,d)) + (integral (g1,c,d))
by A1, A2, A3, A4, A5, Th27
.=
(integral (f,c,d)) + (integral (g,c,d))
by A7, REAL_NS1:2
;
verum