let a, b, c, d be Real; :: 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 ;
then A9: a <= d by ;
A10: d <= b by ;
then ['a,d'] c= ['a,b'] by ;
then A11: ['a,d'] c= dom f by A4;
a <= c by ;
then A12: c in ['a,d'] by ;
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