let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,ExtREAL
for A, E being Element of S st E = dom f & f is E -measurable & f is nonpositive holds
0 >= Integral (M,(f | A))

let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,ExtREAL
for A, E being Element of S st E = dom f & f is E -measurable & f is nonpositive holds
0 >= Integral (M,(f | A))

let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,ExtREAL
for A, E being Element of S st E = dom f & f is E -measurable & f is nonpositive holds
0 >= Integral (M,(f | A))

let f be PartFunc of X,ExtREAL; :: thesis: for A, E being Element of S st E = dom f & f is E -measurable & f is nonpositive holds
0 >= Integral (M,(f | A))

let A, E be Element of S; :: thesis: ( E = dom f & f is E -measurable & f is nonpositive implies 0 >= Integral (M,(f | A)) )
assume that
A1: E = dom f and
A2: f is E -measurable and
A3: f is nonpositive ; :: thesis: 0 >= Integral (M,(f | A))
reconsider E1 = E /\ A as Element of S ;
A4: dom (f | A) = E1 by ;
A5: E1 = (dom f) /\ E1 by ;
f is E1 -measurable by ;
then A6: f | E1 is E1 -measurable by ;
f | E1 = (f | E) | A by RELAT_1:71;
hence 0 >= Integral (M,(f | A)) by A1, A3, A4, A6, Th61, Th1; :: thesis: verum