let f, g be PartFunc of REAL,REAL; :: thesis: for A being non empty closed_interval Subset of REAL st (f (#) g) | A is bounded & f (#) g is_integrable_on A & A c= dom (f (#) g) holds
|||((- f),g,A)||| = - |||(f,g,A)|||
let A be non empty closed_interval Subset of REAL; :: thesis: ( (f (#) g) | A is bounded & f (#) g is_integrable_on A & A c= dom (f (#) g) implies |||((- f),g,A)||| = - |||(f,g,A)||| )
assume A1: ( (f (#) g) | A is bounded & f (#) g is_integrable_on A & A c= dom (f (#) g) ) ; :: thesis: |||((- f),g,A)||| = - |||(f,g,A)|||
|||((- f),g,A)||| = integral (((- 1) (#) (f (#) g)),A) by RFUNCT_1:12
.= (- 1) * (integral ((f (#) g),A)) by A1, INTEGRA6:9
.= - |||(f,g,A)||| ;
hence |||((- f),g,A)||| = - |||(f,g,A)||| ; :: thesis: verum