reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,ExtREAL,
  E for Element of S;

theorem Th2:
  for Y be set, f be PartFunc of X,ExtREAL, r be Real holds (r(#)f)
  |Y = r(#)(f|Y)
proof
  let Y be set, f be PartFunc of X,ExtREAL, r be Real;
A1: now
    let x be Element of X;
    assume
A2: x in dom ((r(#)f)|Y);
    then
A3: x in dom (r(#)f) /\ Y by RELAT_1:61;
    then
A4: x in Y by XBOOLE_0:def 4;
A5: x in dom (r(#)f) by A3,XBOOLE_0:def 4;
    then x in dom f by MESFUNC1:def 6;
    then x in dom f /\ Y by A4,XBOOLE_0:def 4;
    then
A6: x in dom (f|Y) by RELAT_1:61;
    then
A7: x in dom (r(#)(f|Y)) by MESFUNC1:def 6;
    thus ((r(#)f)|Y).x = (r(#)f).x by A2,FUNCT_1:47
      .= (r)*((f.x)) by A5,MESFUNC1:def 6
      .= (r)*((f|Y).x) by A6,FUNCT_1:47
      .= (r(#)(f|Y)).x by A7,MESFUNC1:def 6;
  end;
  dom ((r(#)f)|Y) = dom (r(#)f) /\ Y by RELAT_1:61
    .= dom f /\ Y by MESFUNC1:def 6
    .= dom (f|Y) by RELAT_1:61
    .= dom (r(#)(f|Y)) by MESFUNC1:def 6;
  hence thesis by A1,PARTFUN1:5;
end;
