
theorem Th42:
  for X be non empty set, S be SigmaField of X, M be sigma_Measure
of S, f be PartFunc of X,ExtREAL, B,BF be Element of S st f is B-measurable
  & BF = dom f /\ B holds f|B is BF-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, B,BF be Element of S such that
A1: f is B-measurable and
A2: BF = dom f /\ B;
  now
    let r be Real;
A3: now
      let x be object;
      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;
    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;
  end;
  hence thesis by MESFUNC1:def 16;
end;
