let X be non empty set ; 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 A -measurable & g is A -measurable & A c= dom g holds
(A /\ (less_dom (f,r))) /\ (great_dom (g,r)) in S
let S be SigmaField of X; for f, g being PartFunc of X,ExtREAL
for A being Element of S
for r being Real st f is A -measurable & g is A -measurable & A c= dom g holds
(A /\ (less_dom (f,r))) /\ (great_dom (g,r)) in S
let f, g be PartFunc of X,ExtREAL; for A being Element of S
for r being Real st f is A -measurable & g is A -measurable & A c= dom g holds
(A /\ (less_dom (f,r))) /\ (great_dom (g,r)) in S
let A be Element of S; for r being Real st f is A -measurable & g is A -measurable & A c= dom g holds
(A /\ (less_dom (f,r))) /\ (great_dom (g,r)) in S
let r be Real; ( f is A -measurable & g is A -measurable & A c= dom g implies (A /\ (less_dom (f,r))) /\ (great_dom (g,r)) in S )
assume
( f is A -measurable & g is A -measurable & A c= dom g )
; (A /\ (less_dom (f,r))) /\ (great_dom (g,r)) in S
then A1:
( A /\ (less_dom (f,r)) in S & A /\ (great_dom (g,r)) in S )
by Th29;
(A /\ (less_dom (f,r))) /\ (A /\ (great_dom (g,r))) =
((A /\ (less_dom (f,r))) /\ A) /\ (great_dom (g,r))
by XBOOLE_1:16
.=
((A /\ A) /\ (less_dom (f,r))) /\ (great_dom (g,r))
by XBOOLE_1:16
.=
(A /\ (less_dom (f,r))) /\ (great_dom (g,r))
;
hence
(A /\ (less_dom (f,r))) /\ (great_dom (g,r)) in S
by A1, FINSUB_1:def 2; verum