let X be RealNormSpace; :: thesis: for f being PartFunc of REAL, the carrier of X
for A being non empty closed_interval Subset of REAL
for a, b being Real st A = [.b,a.] & A c= dom f holds
- (integral (f,A)) = integral (f,a,b)
let f be PartFunc of REAL, the carrier of X; :: thesis: for A being non empty closed_interval Subset of REAL
for a, b being Real st A = [.b,a.] & A c= dom f holds
- (integral (f,A)) = integral (f,a,b)
let A be non empty closed_interval Subset of REAL; :: thesis: for a, b being Real st A = [.b,a.] & A c= dom f holds
- (integral (f,A)) = integral (f,a,b)
let a, b be Real; :: thesis: ( A = [.b,a.] & A c= dom f implies - (integral (f,A)) = integral (f,a,b) )
consider a1, b1 being Real such that
A1: a1 <= b1 and
A2: A = [.a1,b1.] by MEASURE5:14;
assume A3: ( A = [.b,a.] & A c= dom f ) ; :: thesis: - (integral (f,A)) = integral (f,a,b)
then A4: ( a1 = b & b1 = a ) by A2, INTEGRA1:5;

now :: thesis: - (integral (f,A)) = integral (f,a,b)

per cases ( b < a or b = a ) by A1, A4, XXREAL_0:1;

suppose A5: b < a ; :: thesis: - (integral (f,A)) = integral (f,a,b)

then integral (f,a,b) = - (integral (f,['b,a'])) by Def9;
hence - (integral (f,A)) = integral (f,a,b) by A3, A5, INTEGRA5:def 3; :: thesis: verum

end;

suppose A6: b = a ; :: thesis: - (integral (f,A)) = integral (f,a,b)

A = [.(lower_bound A),(upper_bound A).] by INTEGRA1:4;
then ( lower_bound A = b & upper_bound A = a ) by A3, INTEGRA1:5;
then A7: vol A = (upper_bound A) - (upper_bound A) by A6
.= 0 ;
A8: integral (f,a,b) = integral (f,A) by A3, A6, Th16;
A9: - (integral (f,a,b)) = - (integral (f,A)) by A3, A6, Th16;
integral (f,a,b) = 0. X by A7, A3, Th17, A8;
hence - (integral (f,A)) = integral (f,a,b) by A9, RLVECT_1:12; :: thesis: verum

end;

end;

end;

hence - (integral (f,A)) = integral (f,a,b) ; :: thesis: verum