let f be PartFunc of REAL,COMPLEX; :: thesis: for A being closed-interval Subset of REAL
for a, b being Real st A = [.b,a.] holds
- (integral (f,A)) = integral (f,a,b)
let A be closed-interval Subset of REAL; :: thesis: for a, b being Real st A = [.b,a.] holds
- (integral (f,A)) = integral (f,a,b)
let a, b be Real; :: thesis: ( A = [.b,a.] implies - (integral (f,A)) = integral (f,a,b) )
assume A1: A = [.b,a.] ; :: thesis: - (integral (f,A)) = integral (f,a,b)
A2: ( Re (integral (f,a,b)) = integral ((Re f),a,b) & Im (integral (f,a,b)) = integral ((Im f),a,b) ) by COMPLEX1:28;
X1: Re (- (integral (f,A))) = - (Re (integral (f,A))) by COMPLEX1:34
.= - (integral ((Re f),A)) by COMPLEX1:28 ;
X2: Im (- (integral (f,A))) = - (Im (integral (f,A))) by COMPLEX1:34
.= - (integral ((Im f),A)) by COMPLEX1:28 ;
Y1: Re (- (integral (f,A))) = Re (integral (f,a,b)) by X1, A1, A2, INTEGRA5:20;
Im (- (integral (f,A))) = Im (integral (f,a,b)) by X2, A1, A2, INTEGRA5:20;
hence - (integral (f,A)) = integral (f,a,b) by COMPLEX1:def 5, Y1; :: thesis: verum