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 that
A1: a <= b and
A2: c <= d and
A3: ( f is_integrable_on ['a,b'] & f | ['a,b'] is bounded ) and
A4: ['a,b'] c= dom f and
A5: c in ['a,b'] and
A6: d in ['a,b'] ; :: thesis: integral f,a,d = (integral f,a,c) + (integral f,c,d)
A7: f is_integrable_on ['a,d'] by A1, A3, A4, A6, Th17;
A8: ['a,b'] = [.a,b.] by A1, INTEGRA5:def 4;
then A9: a <= d by A6, XXREAL_1:1;
A10: d <= b by A6, A8, XXREAL_1:1;
then ['a,d'] c= ['a,b'] by A9, Lm3;
then A11: ['a,d'] c= dom f by A4, XBOOLE_1:1;
a <= c by A5, A8, XXREAL_1:1;
then A12: c in ['a,d'] by A2, Lm3;
f | ['a,d'] is bounded by A3, A4, A9, A10, Th18;
hence integral f,a,d = (integral f,a,c) + (integral f,c,d) by A9, A12, A11, A7, Th17; :: thesis: verum