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

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

let f be PartFunc of REAL,(); :: thesis: ( a <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] implies integral (f,a,d) = (integral (f,a,c)) + (integral (f,c,d)) )
assume A1: ( a <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] ) ; :: thesis: integral (f,a,d) = (integral (f,a,c)) + (integral (f,c,d))
['a,b'] = [.a,b.] by ;
then A2: ( a in ['a,b'] & b in ['a,b'] ) by A1;
reconsider f1 = f as PartFunc of REAL,(REAL n) by REAL_NS1:def 4;
A3: f1 | ['a,b'] is bounded by ;
A4: f1 is_integrable_on ['a,b'] by Th43, A3, A1;
A5: integral (f1,a,d) = (integral (f1,a,c)) + (integral (f1,c,d)) by A3, A4, A1, Th31;
A6: integral (f,a,d) = integral (f1,a,d) by A2, Th48, A1;
A7: integral (f,a,c) = integral (f1,a,c) by A2, Th48, A1;
integral (f,c,d) = integral (f1,c,d) by ;
hence integral (f,a,d) = (integral (f,a,c)) + (integral (f,c,d)) by ; :: thesis: verum