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_measurable_on B & BF = (dom f) /\ B holds
f | B is_measurable_on BF
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_measurable_on B & BF = (dom f) /\ B holds
f | B is_measurable_on BF
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_measurable_on B & BF = (dom f) /\ B holds
f | B is_measurable_on BF
let f be PartFunc of X,ExtREAL ; :: thesis: for B, BF being Element of S st f is_measurable_on B & BF = (dom f) /\ B holds
f | B is_measurable_on BF
let B, BF be Element of S; :: thesis: ( f is_measurable_on B & BF = (dom f) /\ B implies f | B is_measurable_on BF )
assume that
A1: f is_measurable_on B and
A2: BF = (dom f) /\ B ; :: thesis: f | B is_measurable_on BF

now

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

A3: now

let x be set ; :: thesis: ( x in BF /\ (less_dom (f | B),(R_EAL r)) iff x in B /\ (less_dom f,(R_EAL r)) )
( x in dom (f | B) & ex y being R_eal st
( y = (f | B) . x & y < R_EAL r ) iff ( x in (dom f) /\ B & ex y being R_eal st
( y = (f | B) . x & y < R_EAL r ) ) ) by RELAT_1:90;
then A4: ( x in BF & x in less_dom (f | B),(R_EAL r) iff ( x in B & x in dom f & (f | B) . x < R_EAL r ) ) by A2, MESFUNC1:def 12, XBOOLE_0:def 4;
( x in B & x in dom f implies ( f . x < R_EAL r iff (f | B) . x < R_EAL r ) ) by FUNCT_1:72;
then ( x in BF /\ (less_dom (f | B),(R_EAL r)) iff ( x in B & x in less_dom f,(R_EAL r) ) ) by A4, MESFUNC1:def 12, XBOOLE_0:def 4;
hence ( x in BF /\ (less_dom (f | B),(R_EAL r)) iff x in B /\ (less_dom f,(R_EAL r)) ) by XBOOLE_0:def 4; :: thesis: verum

end;

then A5: B /\ (less_dom f,(R_EAL r)) c= BF /\ (less_dom (f | B),(R_EAL r)) by TARSKI:def 3;
BF /\ (less_dom (f | B),(R_EAL r)) c= B /\ (less_dom f,(R_EAL r)) by A3, TARSKI:def 3;
then BF /\ (less_dom (f | B),(R_EAL r)) = B /\ (less_dom f,(R_EAL r)) by A5, XBOOLE_0:def 10;
hence BF /\ (less_dom (f | B),(R_EAL r)) in S by A1, MESFUNC1:def 17; :: thesis: verum

end;

hence f | B is_measurable_on BF by MESFUNC1:def 17; :: thesis: verum