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,
  c for Complex,
  E,A,B for Element of S;

theorem Th41:
  for f be PartFunc of X,REAL,Y,r holds (r(#)f)|Y = r(#)(f|Y)
proof
  let f be PartFunc of X,REAL,Y,r;
A1: dom ((r(#)f)|Y) = dom (r(#)f) /\ Y by RELAT_1:61
    .= dom f /\ Y by VALUED_1:def 5
    .= dom (f|Y) by RELAT_1:61;
A2: dom((r(#)f)|Y) c= dom (r(#)f) by RELAT_1:60;
A3: now
    let x be Element of X;
    assume
A4: x in dom((r(#)f)|Y);
    then x in dom (r(#)(f|Y)) by A1,VALUED_1:def 5;
    then (r(#)(f|Y)).x = r*((f|Y).x) by VALUED_1:def 5;
    then (r(#)(f|Y)).x = r*((f.x)) by A1,A4,FUNCT_1:47;
    then (r(#)(f|Y)).x = (r(#)f).x by A2,A4,VALUED_1:def 5;
    hence (r(#)(f|Y)).x = ((r(#)f)|Y).x by A4,FUNCT_1:47;
  end;
  dom ((r(#)f)|Y) = dom (r(#)(f|Y)) by A1,VALUED_1:def 5;
  hence thesis by A3,PARTFUN1:5;
end;
