
theorem Th48:
  for X be non empty set, S be SigmaField of X, M be sigma_Measure
  of S, f be PartFunc of X,ExtREAL, A,B be Element of S st dom f = A holds f
  is B-measurable iff f is (A/\B)-measurable
proof
  let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be
  PartFunc of X,ExtREAL, A,B be Element of S such that
A1: dom f = A;
A2: now
    let r be Real;
A3: now
      let x be object;
      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;
    end;
    then
A4: less_dom(f, r) c= A /\ less_dom(f, r);
    A /\ less_dom(f, r) c= less_dom(f, r) by A3;
    hence A /\ less_dom(f, r) = less_dom(f, r) by A4;
  end;
  hereby
    assume
A5: f is B-measurable;
    now
      let r be Real;
      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 A5,MESFUNC1:def 16;
    end;
    hence f is (A/\B)-measurable by MESFUNC1:def 16;
  end;
  assume
A6: f is (A/\B)-measurable;
  now
    let r be Real;
    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 A6,MESFUNC1:def 16;
  end;
  hence thesis by MESFUNC1:def 16;
end;
