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 A1, INTEGRA5:def 3;

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;

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

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 3;

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;

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