let a, b, c, d be real number ; :: thesis: for f being PartFunc of REAL ,REAL st a <= b & c <= d & 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 ,REAL ; :: thesis: ( a <= b & c <= d & 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 & c <= d & 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)
then ['a,b'] = [.a,b.] by INTEGRA5:def 4;
then A2: ( a <= d & a <= c & d <= b ) by A1, XXREAL_1:1;
then A3: ( c in ['a,d'] & ['a,d'] c= ['a,b'] ) by A1, Lm3;
then A4: ['a,d'] c= dom f by A1, XBOOLE_1:1;
A5: f is_integrable_on ['a,d'] by A1, Th17;
f | ['a,d'] is bounded by A1, A2, Th18;
hence integral f,a,d = (integral f,a,c) + (integral f,c,d) by A2, A3, A4, A5, Th17; :: thesis: verum