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 = [.a,b.] 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 = [.a,b.] 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 = [.a,b.] holds
integral (f,A) = integral (f,a,b)
let a, b be Real; :: thesis: ( A = [.a,b.] 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 A = [.a,b.] ; :: thesis: integral (f,A) = integral (f,a,b)
then A3: ( a1 = a & b1 = b ) by A2, INTEGRA1:5;
then integral (f,a,b) = integral (f,['a,b']) by A1, Def9;
hence integral (f,A) = integral (f,a,b) by A1, A2, A3, INTEGRA5:def 3; :: thesis: verum