reserve A for set, x,y,z for object,
  k for Element of NAT;
reserve n for Nat,
  x for object;
reserve V, C for set;

theorem
  for f being Element of PFuncs (V, C), g being set st g c= f holds
  g in PFuncs (V, C)
proof
  let f be Element of PFuncs (V, C), g be set;
  consider f1 be Function such that
A1: f1 = f and
A2: dom f1 c= V and
A3: rng f1 c= C by PARTFUN1:def 3;
  assume
A4: g c= f;
  then reconsider g9 = g as Function;
  rng g9 c= rng f1 by A4,A1,RELAT_1:11;
  then
A5: rng g9 c= C by A3;
  dom g9 c= dom f1 by A4,A1,RELAT_1:11;
  then dom g9 c= V by A2;
  hence thesis by A5,PARTFUN1:def 3;
end;
