let X be non empty set ; :: thesis: for S being SigmaField of X
for f being PartFunc of X,REAL
for A being Element of S
for r, s being Real st f is A -measurable & A c= dom f holds
(A /\ (great_eq_dom (f,r))) /\ (less_dom (f,s)) in S
let S be SigmaField of X; :: thesis: for f being PartFunc of X,REAL
for A being Element of S
for r, s being Real st f is A -measurable & A c= dom f holds
(A /\ (great_eq_dom (f,r))) /\ (less_dom (f,s)) in S
let f be PartFunc of X,REAL; :: thesis: for A being Element of S
for r, s being Real st f is A -measurable & A c= dom f holds
(A /\ (great_eq_dom (f,r))) /\ (less_dom (f,s)) in S
let A be Element of S; :: thesis: for r, s being Real st f is A -measurable & A c= dom f holds
(A /\ (great_eq_dom (f,r))) /\ (less_dom (f,s)) in S
let r, s be Real; :: thesis: ( f is A -measurable & A c= dom f implies (A /\ (great_eq_dom (f,r))) /\ (less_dom (f,s)) in S )
assume that
A1: f is A -measurable and
A2: A c= dom f ; :: thesis: (A /\ (great_eq_dom (f,r))) /\ (less_dom (f,s)) in S
R_EAL f is A -measurable by A1;
then A3: A /\ (less_dom ((R_EAL f),s)) in S by MESFUNC1:def 16;
A4: (A /\ (great_eq_dom (f,r))) /\ (A /\ (less_dom (f,s))) = ((A /\ (great_eq_dom (f,r))) /\ A) /\ (less_dom (f,s)) by XBOOLE_1:16
.= ((great_eq_dom (f,r)) /\ (A /\ A)) /\ (less_dom (f,s)) by XBOOLE_1:16 ;
A /\ (great_eq_dom (f,r)) in S by A1, A2, Th13;
hence (A /\ (great_eq_dom (f,r))) /\ (less_dom (f,s)) in S by A3, A4, FINSUB_1:def 2; :: thesis: verum