reserve X for set;
reserve X,X1,X2 for non empty set;
reserve S for SigmaField of X;
reserve S1 for SigmaField of X1;
reserve S2 for SigmaField of X2;
reserve M for sigma_Measure of S;
reserve M1 for sigma_Measure of S1;
reserve M2 for sigma_Measure of S2;

theorem
for A,B being Element of S, f being PartFunc of X,ExtREAL
st B c= A & f|A is B-measurable holds f is B-measurable
proof
    let A,B be Element of S, f be PartFunc of X,ExtREAL;
    assume that
A1:  B c= A and
A2:  f|A is B-measurable;
     let r be Real;
     now let x be object;
      assume x in B /\ less_dom(f|A,r); then
A3:   x in B & x in less_dom(f|A,r) by XBOOLE_0:def 4; then
      x in dom(f|A) & (f|A).x < r by MESFUNC1:def 11; then
      x in dom f & f.x < r by RELAT_1:57,FUNCT_1:47; then
      x in less_dom(f,r) by MESFUNC1:def 11;
      hence x in B /\ less_dom(f,r) by A3,XBOOLE_0:def 4;
     end; then
A4:  B /\ less_dom(f|A,r) c= B /\ less_dom(f,r);
     now let x be object;
      assume x in B /\ less_dom(f,r); then
A5:   x in B & x in less_dom(f,r) by XBOOLE_0:def 4; then
      x in dom f & f.x < r by MESFUNC1:def 11; then
      x in dom(f|A) & (f|A).x < r by A1,A5,RELAT_1:57,FUNCT_1:49; then
      x in less_dom(f|A,r) by MESFUNC1:def 11;
      hence x in B /\ less_dom(f|A,r) by A5,XBOOLE_0:def 4;
     end; then
     B /\ less_dom(f,r) c= B /\ less_dom(f|A,r); then
     B /\ less_dom(f|A,r) = B /\ less_dom(f,r) by A4;
     hence B /\ less_dom(f,r) in S by A2;
end;
