theorem Th4:
  dom (r(#)f) = dom f &
  for c st c in dom (r(#)f) holds (r(#)f)/.c = r * (f/.c)
proof
  thus
A1: dom (r(#)f) = dom f by VALUED_1:def 5;
  now
    let c;
    assume
A2: c in dom (r(#)f);
    hence (r(#)f)/.c = (r(#)f).c by PARTFUN1:def 6
      .= r *(f.c) by VALUED_1:6
      .= r *(f/.c) by A1,A2,PARTFUN1:def 6;
  end;
  hence thesis;
end;
