let X be non empty set ; :: thesis:
let S be SigmaField of X; :: thesis:
let M be sigma_Measure of S; :: thesis:
let f be PartFunc of X,ExtREAL; :: thesis:
let A, B be Element of S; :: thesis:
assume A1: dom f = A ; :: thesis:

A2: now :: thesis:

let r be Real; :: thesis:

A3: now :: thesis:

let x be object ; :: thesis:
( x in A /\ (less_dom (f,r)) iff ( x in A & x in less_dom (f,r) ) ) by XBOOLE_0:def 4;
hence ( x in A /\ (less_dom (f,r)) iff x in less_dom (f,r) ) by A1, MESFUNC1:def 11; :: thesis:

end;

then A4: less_dom (f,r) c= A /\ (less_dom (f,r)) ;
A /\ (less_dom (f,r)) c= less_dom (f,r) by A3;
hence A /\ (less_dom (f,r)) = less_dom (f,r) by A4; :: thesis:

end;

hereby :: thesis:

assume A5: f is B -measurable ; :: thesis:

now :: thesis:

let r be Real; :: thesis:
(A /\ B) /\ (less_dom (f,r)) = B /\ (A /\ (less_dom (f,r))) by XBOOLE_1:16
.= B /\ (less_dom (f,r)) by A2 ;
hence (A /\ B) /\ (less_dom (f,r)) in S by A5, MESFUNC1:def 16; :: thesis:

end;

hence f is A /\ B -measurable by MESFUNC1:def 16; :: thesis:

end;

assume A6: f is A /\ B -measurable ; :: thesis:

now :: thesis:

let r be Real; :: thesis:
(A /\ B) /\ (less_dom (f,r)) = B /\ (A /\ (less_dom (f,r))) by XBOOLE_1:16
.= B /\ (less_dom (f,r)) by A2 ;
hence B /\ (less_dom (f,r)) in S by A6, MESFUNC1:def 16; :: thesis:

end;

hence f is B -measurable by MESFUNC1:def 16; :: thesis: