
theorem Th33:
  for R being connected non empty Poset,
  x being Element of Fin the carrier of R
  st x <> {} holds not [x,{}] in union rng FinOrd-Approx R
proof
  let R be connected non empty Poset,
  x be Element of Fin the carrier of R such that
A1: x <> {};
  set CR = the carrier of R, FOAR = FinOrd-Approx R;
  reconsider y={} as Element of Fin CR by FINSUB_1:7;
  now
    assume
A2: [x,y] in union rng FinOrd-Approx R;
    per cases by A2,Th32;
    suppose x = {};
      hence contradiction by A1;
    end;
    suppose x<>{} & y<>{} & [PosetMax x,PosetMax y] in CR;
      hence contradiction;
    end;
    suppose x<>{} & y<>{} & PosetMax x = PosetMax y &
      [x\PosetMax x, {}\PosetMax y] in union rng FOAR;
      hence contradiction;
    end;
  end;
  hence thesis;
end;
