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 Th15:
  for A, B being Element of Fin PFuncs (V, C), s being set st s in
  A =>> B holds ex f being PartFunc of A, B st s = union {f.i \ i where i is
  Element of PFuncs (V, C) : i in A} & dom f = A
proof
  let A, B be Element of Fin PFuncs (V, C), s be set;
  assume s in A =>> B;
  then s in { union {f.i \ i where i is Element of PFuncs (V, C) : i in A}
  where f is Element of PFuncs (A, B) : dom f = A} by XBOOLE_0:def 4;
  then consider f be Element of PFuncs (A, B) such that
A1: s = union {f.i \ i where i is Element of PFuncs (V, C) : i in A} and
A2: dom f = A;
  f is PartFunc of A, B by PARTFUN1:47;
  hence thesis by A1,A2;
end;
