theorem Th51:
  for A being non empty set for f being Function of A, Fin A holds
  f = singleton A iff for x being Element of A holds f.x = {x}
proof
  let A be non empty set;
  let f be Function of A, Fin A;
  thus f = singleton A implies for x being Element of A holds f.x = {x} by Def6
;
  assume for x being Element of A holds f.x = {x};
  then for x be object holds x in A implies f.x = {x};
  hence thesis by Def6;
end;
