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 Th7:
  for V being set, C being finite set, A being Element of Fin
  PFuncs (V, C) st A = {} holds Involved A = {}
proof
  let V be set, C be finite set, A be Element of Fin PFuncs (V, C);
  assume
A1: A = {};
  assume Involved A <> {};
  then consider x be object such that
A2: x in Involved A by XBOOLE_0:def 1;
  ex f being finite Function st f in A & x in dom f by A2,Def1;
  hence thesis by A1;
end;
