let X be non empty set ; :: thesis: for S being SigmaField of X
for f, g being PartFunc of X,REAL
for A being Element of S
for r being Real st f is_measurable_on A & g is_measurable_on A & A c= dom g holds
(A /\ (less_dom (f,r))) /\ (great_dom (g,r)) in S
let S be SigmaField of X; :: thesis: for f, g being PartFunc of X,REAL
for A being Element of S
for r being Real st f is_measurable_on A & g is_measurable_on A & A c= dom g holds
(A /\ (less_dom (f,r))) /\ (great_dom (g,r)) in S
let f, g be PartFunc of X,REAL; :: thesis: for A being Element of S
for r being Real st f is_measurable_on A & g is_measurable_on A & A c= dom g holds
(A /\ (less_dom (f,r))) /\ (great_dom (g,r)) in S
let A be Element of S; :: thesis: for r being Real st f is_measurable_on A & g is_measurable_on A & A c= dom g holds
(A /\ (less_dom (f,r))) /\ (great_dom (g,r)) in S
let r be Real; :: thesis: ( f is_measurable_on A & g is_measurable_on A & A c= dom g implies (A /\ (less_dom (f,r))) /\ (great_dom (g,r)) in S )
assume that
A1: ( f is_measurable_on A & g is_measurable_on A ) and
A2: A c= dom g ; :: thesis: (A /\ (less_dom (f,r))) /\ (great_dom (g,r)) in S
( R_EAL f is_measurable_on A & R_EAL g is_measurable_on A ) by A1, Def6;
then (A /\ (less_dom (f,(R_EAL r)))) /\ (great_dom (g,(R_EAL r))) in S by A2, MESFUNC1:36;
hence (A /\ (less_dom (f,r))) /\ (great_dom (g,r)) in S ; :: thesis: verum