let X be non empty set ; :: thesis: for S being SigmaField of X
for f being PartFunc of X,REAL
for A, B being Element of S st dom f = A holds
( f is B -measurable iff f is A /\ B -measurable )
let S be SigmaField of X; :: thesis: for f being PartFunc of X,REAL
for A, B being Element of S st dom f = A holds
( f is B -measurable iff f is A /\ B -measurable )
let f be PartFunc of X,REAL; :: thesis: for A, B being Element of S st dom f = A holds
( f is B -measurable iff f is A /\ B -measurable )
let A, B be Element of S; :: thesis: ( dom f = A implies ( f is B -measurable iff f is A /\ B -measurable ) )
assume A1: dom f = A ; :: thesis: ( f is B -measurable iff f is A /\ B -measurable )

A2: now :: thesis: for r being Real holds A /\ (less_dom (f,r)) = less_dom (f,r)

let r be Real; :: thesis: A /\ (less_dom (f,r)) = less_dom (f,r)

now :: thesis: for x being object holds
( x in A /\ (less_dom (f,r)) iff x in less_dom (f,r) )

let x be object ; :: thesis: ( x in A /\ (less_dom (f,r)) iff x in less_dom (f,r) )
( 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: verum

end;

then ( A /\ (less_dom (f,r)) c= less_dom (f,r) & less_dom (f,r) c= A /\ (less_dom (f,r)) ) ;
hence A /\ (less_dom (f,r)) = less_dom (f,r) ; :: thesis: verum

end;

hereby :: thesis: ( f is A /\ B -measurable implies f is B -measurable )

assume A3: f is B -measurable ; :: thesis: f is A /\ B -measurable

now :: thesis: for r being Real holds (A /\ B) /\ (less_dom (f,r)) in S

let r be Real; :: thesis: (A /\ B) /\ (less_dom (f,r)) in S
(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 A3, Th12; :: thesis: verum

end;

hence f is A /\ B -measurable by Th12; :: thesis: verum

end;

assume A4: f is A /\ B -measurable ; :: thesis: f is B -measurable

now :: thesis: for r being Real holds B /\ (less_dom (f,r)) in S

let r be Real; :: thesis: B /\ (less_dom (f,r)) in S
(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 A4, Th12; :: thesis: verum

end;

hence f is B -measurable by Th12; :: thesis: verum