reserve Z for set;

theorem
  for n being Nat, Z,x being set, f being PartFunc of Z,REAL n
  for r being Real
  st x in dom (r(#)f) holds (r(#)f)/.x = r* (f/.x)
  proof
    let n be Nat;
    let Z,x be set;
    let f be PartFunc of Z,REAL n;
    let r be Real;
    assume
A1: x in dom (r(#)f);
    dom (r(#)f) = dom f by VALUED_2:def 39;
    then
A2: f.x = f/.x by A1,PARTFUN1:def 6;
    thus (r(#)f)/.x = (r(#)f).x by A1,PARTFUN1:def 6
    .= r * (f/.x) by A1,A2,VALUED_2:def 39;
  end;
