
theorem Th4: :: uogolnione WAYBEL_3:16
  for T being with_suprema Poset,
  D being non empty directed finite Subset of T holds sup D in D
proof
  let T be reflexive transitive antisymmetric with_suprema RelStr,
  D be non empty directed finite Subset of T;
  D c= D;
  then reconsider E = D as finite Subset of D;
  consider x being Element of T such that
A1: x in D and
A2: x is_>=_than E by WAYBEL_0:1;
A3: for b being Element of T st D is_<=_than
  b holds b >= x by A1;
  for c being Element of T st D is_<=_than c &
  for b being Element of T st D is_<=_than b holds b >= c holds c = x
  proof
    let c be Element of T such that
A4: D is_<=_than c and
A5: for b being Element of T st D is_<=_than b holds b >= c;
A6: x >= c by A2,A5;
    c >= x by A1,A4;
    hence thesis by A6,ORDERS_2:2;
  end;
  then ex_sup_of D,T by A2,A3;
  hence thesis by A1,A2,A3,YELLOW_0:def 9;
end;
