let X be non empty set ; :: thesis: for S being SigmaField of X
for f, g being PartFunc of X,ExtREAL
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_EAL r))) /\ (great_dom g,(R_EAL r)) in S
let S be SigmaField of X; :: thesis: for f, g being PartFunc of X,ExtREAL
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_EAL r))) /\ (great_dom g,(R_EAL r)) in S
let f, g be PartFunc of X,ExtREAL ; :: 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_EAL r))) /\ (great_dom g,(R_EAL 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_EAL r))) /\ (great_dom g,(R_EAL 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_EAL r))) /\ (great_dom g,(R_EAL r)) in S )
assume A1: ( f is_measurable_on A & g is_measurable_on A & A c= dom g ) ; :: thesis: (A /\ (less_dom f,(R_EAL r))) /\ (great_dom g,(R_EAL r)) in S
then A2: A /\ (less_dom f,(R_EAL r)) in S by Def17;
A3: A /\ (great_dom g,(R_EAL r)) in S by A1, Th33;
(A /\ (less_dom f,(R_EAL r))) /\ (A /\ (great_dom g,(R_EAL r))) = ((A /\ (less_dom f,(R_EAL r))) /\ A) /\ (great_dom g,(R_EAL r)) by XBOOLE_1:16
.= ((A /\ A) /\ (less_dom f,(R_EAL r))) /\ (great_dom g,(R_EAL r)) by XBOOLE_1:16
.= (A /\ (less_dom f,(R_EAL r))) /\ (great_dom g,(R_EAL r)) ;
hence (A /\ (less_dom f,(R_EAL r))) /\ (great_dom g,(R_EAL r)) in S by A2, A3, MEASURE1:19; :: thesis: verum