
theorem
  for T being finite up-complete non empty Poset,
  S being Subset of T holds S is inaccessible
proof
  let T be finite up-complete non empty Poset,
  S be Subset of T, D be non empty directed Subset of T such that
A1: sup D in S;
  sup D in D
  proof
    reconsider D9 = D as finite Subset of T;
    D9 c= D9;
    then reconsider E = D9 as finite Subset of D;
    consider x being Element of T such that
A2: x in D and
A3: x is_>=_than E by WAYBEL_0:1;
A4: for b being Element of T st D is_<=_than b holds b >= x by A2;
    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
A5:   D is_<=_than c and
A6:   for b being Element of T st D is_<=_than b holds b >= c;
A7:   x >= c by A3,A6;
      c >= x by A2,A5;
      hence thesis by A7,ORDERS_2:2;
    end;
    then ex_sup_of D,T by A3,A4,YELLOW_0:def 7;
    hence thesis by A2,A3,A4,YELLOW_0:def 9;
  end;
  hence thesis by A1,XBOOLE_0:3;
end;
