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

theorem Th2:
  for a being finite Element of PFuncs (V, C) holds {a} in
  SubstitutionSet (V, C)
proof
  let a be finite Element of PFuncs (V, C);
A1: for s, t being Element of PFuncs (V,C) holds ( s in { a } & t in { a } &
  s c= t implies s = t )
  proof
    let s, t be Element of PFuncs (V,C);
    assume that
A2: s in { a } and
A3: t in { a } and
    s c= t;
    s = a by A2,TARSKI:def 1;
    hence thesis by A3,TARSKI:def 1;
  end;
  for u being set st u in {a} holds u is finite;
  then
  {.a.} in { A where A is Element of Fin PFuncs (V,C) : ( for u being set
st u in A holds u is finite ) & for s, t being Element of PFuncs (V, C) holds (
  s in A & t in A & s c= t implies s = t ) } by A1;
  hence thesis by SUBSTLAT:def 1;
end;
