let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL st ex E being Element of S st
( E = dom f & E = dom g & f is E -measurable & g is E -measurable ) & f is nonnegative & g is nonnegative & ( for x being Element of X st x in dom g holds
g . x <= f . x ) holds
Integral (M,g) <= Integral (M,f)

let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,REAL st ex E being Element of S st
( E = dom f & E = dom g & f is E -measurable & g is E -measurable ) & f is nonnegative & g is nonnegative & ( for x being Element of X st x in dom g holds
g . x <= f . x ) holds
Integral (M,g) <= Integral (M,f)

let M be sigma_Measure of S; :: thesis: for f, g being PartFunc of X,REAL st ex E being Element of S st
( E = dom f & E = dom g & f is E -measurable & g is E -measurable ) & f is nonnegative & g is nonnegative & ( for x being Element of X st x in dom g holds
g . x <= f . x ) holds
Integral (M,g) <= Integral (M,f)

let f, g be PartFunc of X,REAL; :: thesis: ( ex E being Element of S st
( E = dom f & E = dom g & f is E -measurable & g is E -measurable ) & f is nonnegative & g is nonnegative & ( for x being Element of X st x in dom g holds
g . x <= f . x ) implies Integral (M,g) <= Integral (M,f) )

assume that
A1: ex A being Element of S st
( A = dom f & A = dom g & f is A -measurable & g is A -measurable ) and
A2: ( f is nonnegative & g is nonnegative ) and
A3: for x being Element of X st x in dom g holds
g . x <= f . x ; :: thesis: Integral (M,g) <= Integral (M,f)
A4: ( Integral (M,g) = integral+ (M,()) & Integral (M,f) = integral+ (M,()) ) by ;
consider A being Element of S such that
A5: ( A = dom f & A = dom g ) and
A6: ( f is A -measurable & g is A -measurable ) by A1;
( R_EAL f is A -measurable & R_EAL g is A -measurable ) by ;
hence Integral (M,g) <= Integral (M,f) by ; :: thesis: verum