reserve V, C, x, a, b for set;
reserve A, B for Element of SubstitutionSet (V, C);
reserve C for finite set;
reserve A, B for Element of SubstitutionSet (V, C);

theorem Th6:
  for V being set, C be finite set, A being Element of Fin PFuncs
  (V, C) holds Involved A c= V
proof
  let V be set, C be finite set, A be Element of Fin PFuncs (V, C);
  let a be object;
  assume a in Involved A;
  then consider f being finite Function such that
A1: f in A and
A2: a in dom f by Def1;
  A c= PFuncs (V, C) by FINSUB_1:def 5;
  then ex f1 being Function st f = f1 & dom f1 c= V & rng f1 c= C by A1,
PARTFUN1:def 3;
  hence thesis by A2;
end;
