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 A being Element of S st
( A = (dom f) /\ (dom g) & f is A -measurable & g is A -measurable ) & f is_integrable_on M & g is_integrable_on M & g - f is nonnegative holds
ex E being Element of S st
( E = (dom f) /\ (dom g) & Integral (M,(f | E)) <= Integral (M,(g | E)) )

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

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

let f, g be PartFunc of X,REAL; :: thesis: ( ex A being Element of S st
( A = (dom f) /\ (dom g) & f is A -measurable & g is A -measurable ) & f is_integrable_on M & g is_integrable_on M & g - f is nonnegative implies ex E being Element of S st
( E = (dom f) /\ (dom g) & Integral (M,(f | E)) <= Integral (M,(g | E)) ) )

assume that
A1: ex A being Element of S st
( A = (dom f) /\ (dom g) & f is A -measurable & g is A -measurable ) and
A2: f is_integrable_on M and
A3: g is_integrable_on M and
A4: g - f is nonnegative ; :: thesis: ex E being Element of S st
( E = (dom f) /\ (dom g) & Integral (M,(f | E)) <= Integral (M,(g | E)) )

set h = (- 1) (#) f;
(- 1) (#) f is_integrable_on M by ;
then consider E being Element of S such that
A5: E = (dom ((- 1) (#) f)) /\ (dom g) and
A6: Integral (M,(((- 1) (#) f) + g)) = (Integral (M,(((- 1) (#) f) | E))) + (Integral (M,(g | E))) by ;
A7: f | E is_integrable_on M by ;
then A8: Integral (M,(f | E)) < +infty by MESFUNC6:90;
-infty < Integral (M,(f | E)) by ;
then reconsider c1 = Integral (M,(f | E)) as Element of REAL by ;
A9: (- 1) * (Integral (M,(f | E))) = (- 1) * c1 by EXTREAL1:1;
A10: g | E is_integrable_on M by ;
then A11: Integral (M,(g | E)) < +infty by MESFUNC6:90;
-infty < Integral (M,(g | E)) by ;
then reconsider c2 = Integral (M,(g | E)) as Element of REAL by ;
take E ; :: thesis: ( E = (dom f) /\ (dom g) & Integral (M,(f | E)) <= Integral (M,(g | E)) )
A12: ((- 1) (#) f) | E = (- 1) (#) (f | E) by Th41;
consider A being Element of S such that
A13: A = (dom f) /\ (dom g) and
A14: f is A -measurable and
A15: g is A -measurable by A1;
dom ((- 1) (#) f) = dom f by VALUED_1:def 5;
then A16: dom (((- 1) (#) f) + g) = A by ;
(- 1) (#) f is A -measurable by ;
then 0 <= (Integral (M,(((- 1) (#) f) | E))) + (Integral (M,(g | E))) by ;
then 0 <= ((- 1) * (Integral (M,(f | E)))) + (Integral (M,(g | E))) by ;
then 0 <= (- c1) + c2 by ;
then 0 + c1 <= ((- c1) + c2) + c1 by XREAL_1:6;
hence ( E = (dom f) /\ (dom g) & Integral (M,(f | E)) <= Integral (M,(g | E)) ) by ; :: thesis: verum