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 ( 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 A1: ( A /\ (less_dom (f,(R_EAL r))) in S & A /\ (great_dom (g,(R_EAL r))) in S ) by Def17, 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 A1, FINSUB_1:def 2; :: thesis: verum