theorem
  for X,S,f,A,B st B c= A & f is A-measurable holds f is B-measurable
proof
  let X,S,f,A,B;
  assume that
A1: B c= A and
A2: f is A-measurable;
 for r being Real holds (B /\ less_dom(f,r)) in S
  proof
    let r be Real;
A3: A /\ less_dom(f,r) in S by A2;
 B /\ (A /\ less_dom(f,r)) = B /\ A /\ less_dom(f,r) by XBOOLE_1:16
      .= B /\ less_dom(f,r) by A1,XBOOLE_1:28;
    hence thesis by A3,FINSUB_1:def 2;
  end;
  hence thesis;
end;
