let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,ExtREAL
for B, BF being Element of S st f is B -measurable & BF = (dom f) /\ B holds
f | B is BF -measurable
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,ExtREAL
for B, BF being Element of S st f is B -measurable & BF = (dom f) /\ B holds
f | B is BF -measurable
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,ExtREAL
for B, BF being Element of S st f is B -measurable & BF = (dom f) /\ B holds
f | B is BF -measurable
let f be PartFunc of X,ExtREAL; :: thesis: for B, BF being Element of S st f is B -measurable & BF = (dom f) /\ B holds
f | B is BF -measurable
let B, BF be Element of S; :: thesis: ( f is B -measurable & BF = (dom f) /\ B implies f | B is BF -measurable )
assume that
A1: f is B -measurable and
A2: BF = (dom f) /\ B ; :: thesis: f | B is BF -measurable

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

let r be Real; :: thesis: BF /\ (less_dom ((f | B),r)) in S

A3: now :: thesis: for x being object holds
( x in BF /\ (less_dom ((f | B),r)) iff x in B /\ (less_dom (f,r)) )

let x be object ; :: thesis: ( x in BF /\ (less_dom ((f | B),r)) iff x in B /\ (less_dom (f,r)) )
reconsider xx = x as set by TARSKI:1;
( x in dom (f | B) & ex y being R_eal st
( y = (f | B) . x & y < r ) iff ( x in (dom f) /\ B & ex y being R_eal st
( y = (f | B) . x & y < r ) ) ) by RELAT_1:61;
then A4: ( x in BF & x in less_dom ((f | B),r) iff ( x in B & x in dom f & (f | B) . xx < r ) ) by A2, MESFUNC1:def 11, XBOOLE_0:def 4;
( x in B & x in dom f implies ( f . x < r iff (f | B) . x < r ) ) by FUNCT_1:49;
then ( x in BF /\ (less_dom ((f | B),r)) iff ( x in B & x in less_dom (f,r) ) ) by A4, MESFUNC1:def 11, XBOOLE_0:def 4;
hence ( x in BF /\ (less_dom ((f | B),r)) iff x in B /\ (less_dom (f,r)) ) by XBOOLE_0:def 4; :: thesis: verum

end;

then A5: B /\ (less_dom (f,r)) c= BF /\ (less_dom ((f | B),r)) ;
BF /\ (less_dom ((f | B),r)) c= B /\ (less_dom (f,r)) by A3;
then BF /\ (less_dom ((f | B),r)) = B /\ (less_dom (f,r)) by A5;
hence BF /\ (less_dom ((f | B),r)) in S by A1, MESFUNC1:def 16; :: thesis: verum

end;

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