
theorem
  for L being non empty RelStr, J, x being set for f being Function of J
  ,the carrier of L holds x is Element of FinSups f iff x is Element of Fin J
proof
  let L be non empty RelStr, J, x be set, f be Function of J,the carrier of L;
  ex g being Function of Fin J, the carrier of L st for x being Element of
Fin J holds g.x = sup (f.:x) & FinSups f = NetStr (# Fin J, RelIncl Fin J, g #)
  by Def2;
  hence thesis;
end;
