 reserve A for non empty Subset of REAL;

theorem
  for r be Real, f be PartFunc of REAL,REAL st A c= dom f
    holds
  (r (#) f) | A = r (#) (f | A)
proof
 let r be Real, f be PartFunc of REAL,REAL;
 assume AA: A c= dom f; then
 A1: A c= dom (r (#) f) by VALUED_1:def 5;
 set F = (r (#) f) | A;
 set g = r (#) (f | A);
 A2: dom (r (#) (f | A))
 = dom ((f | A)) by VALUED_1:def 5 .= A by RELAT_1:62,AA;
 for x being object st x in dom F holds F . x = g . x
 proof
  let x be object;
  assume D1:x in dom F; then
  reconsider x as Real;
  F . x = (r (#) f).x by FUNCT_1:49,D1
  .= r * (f).x by VALUED_1:6
  .= r * (f | A).x by FUNCT_1:49,D1
  .= g . x by VALUED_1:6;
  hence thesis;
 end;
 hence thesis by FUNCT_1:2,A1,A2,RELAT_1:62;
end;
