reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,COMPLEX,
  r for Real,
  k for Real,
  n for Nat,
  E for Element of S;

theorem
  (r(#)f)|Y = r(#)(f|Y)
proof
A1: dom ((r(#)f)|Y) = dom (r(#)f) /\ Y by RELAT_1:61;
  then dom ((r(#)f)|Y) = dom f /\ Y by VALUED_1:def 5;
  then
A2: dom ((r(#)f)|Y) = dom (f|Y) by RELAT_1:61;
  then
A3: dom ((r(#)f)|Y) = dom (r(#)(f|Y)) by VALUED_1:def 5;
  now
    let x be Element of X;
    assume
A4: x in dom ((r(#)f)|Y);
    then
A5: x in dom (r(#)f) by A1,XBOOLE_0:def 4;
    thus ((r(#)f)|Y).x = (r(#)f).x by A4,FUNCT_1:47
      .= r*(f.x) by A5,VALUED_1:def 5
      .= r*((f|Y).x) by A2,A4,FUNCT_1:47
      .= (r(#)(f|Y)).x by A3,A4,VALUED_1:def 5;
  end;
  hence thesis by A3,PARTFUN1:5;
end;
