
theorem
  for T being up-complete non empty TopPoset holds
  T is upper implies T is order_consistent
proof
  let T be up-complete non empty TopPoset;
  assume
A1: T is upper;
  then reconsider BB = the set of all (downarrow x)` where x is Element of T
 as prebasis of T;
  for x being Element of T holds downarrow x = Cl {x} &
  for N being eventually-directed net of T st x = sup N
  for V being a_neighborhood of x holds N is_eventually_in V
  proof
    let x be Element of T;
A2: downarrow x c= Cl {x}
    proof
      let a be object;
      assume
A3:   a in downarrow x;
      then reconsider a9 = a as Point of T;
      for G being Subset of T st G is open holds a9 in G implies {x} meets G
      proof
        let G be Subset of T such that
A4:     G is open;
        assume
A5:     a9 in G;
        reconsider v = a9 as Element of T;
A6:     v <= x by A3,WAYBEL_0:17;
        G is upper by A1,A4,Th1;
        then
A7:     x in G by A5,A6;
        x in {x} by TARSKI:def 1;
        hence thesis by A7,XBOOLE_0:3;
      end;
      hence thesis by PRE_TOPC:24;
    end;
    Cl {x} c= downarrow x
    proof
      let a be object;
      assume
A8:   a in Cl{x};
      reconsider BB as Subset-Family of T;
   (downarrow x)` in BB;
then A9:  (downarrow x)` is open by TOPS_2:def 1;
      (downarrow x)` = [#]T\downarrow x;
      then
A10:  downarrow x is closed by A9,PRE_TOPC:def 3;
      x <= x;
      then x in downarrow x by WAYBEL_0:17;
      then {x} c= downarrow x by ZFMISC_1:31;
      hence thesis by A8,A10,PRE_TOPC:15;
    end;
    hence downarrow x = Cl {x} by A2;
    thus for N being eventually-directed net of T st x = sup N
    for V being a_neighborhood of x holds N is_eventually_in V
    proof
      let N be eventually-directed net of T;
      assume x = sup N;
      then
A11:  x = Sup the mapping of N by WAYBEL_2:def 1
        .= sup rng netmap (N,T) by YELLOW_2:def 5;
      let V be a_neighborhood of x;
A12:  x in Int V by CONNSP_2:def 1;
      FinMeetCl BB is Basis of T by YELLOW_9:23;
      then
A13:  the topology of T = UniCl FinMeetCl BB by YELLOW_9:22;
      Int V in the topology of T by PRE_TOPC:def 2;
      then consider Y being Subset-Family of T such that
A14:  Y c= FinMeetCl BB and
A15:  Int V = union Y by A13,CANTOR_1:def 1;
      consider Y1 being set such that
A16:  x in Y1 and
A17:  Y1 in Y by A12,A15,TARSKI:def 4;
      consider Z being Subset-Family of T such that
A18:  Z c= BB and
A19:  Z is finite and
A20:  Y1 = Intersect Z by A14,A17,CANTOR_1:def 3;
      reconsider rngN = rng netmap (N,T) as Subset of T;
      rngN is directed by WAYBEL_2:18;
      then ex a being Element of T st a is_>=_than rngN &
      for b being Element of T st b is_>=_than rngN holds a <= b
      by WAYBEL_0:def 39;
      then
A21:  ex_sup_of rngN,T by YELLOW_0:15;
      defpred P[object,object] means
      ex D1 being set st D1 = $1 &
      for i,j being Element of N st i = $2 & j >= i holds N.j in D1;
A22:  for Q being object st Q in Z
   ex b being object st b in the carrier of N & P[Q,b]
      proof
        let Q be object;
        assume
A23:    Q in Z;
        then Q in BB by A18;
        then consider v1 being Element of T such that
A24:    Q = (downarrow v1)`;
        x in (downarrow v1)` by A16,A20,A23,A24,SETFAM_1:43;
        then not x in downarrow v1 by XBOOLE_0:def 5;
        then
A25:    not x <= v1 by WAYBEL_0:17;
        not rngN c= downarrow v1
        proof
          assume
A26:      rngN c= downarrow v1;
          ex_sup_of downarrow v1,T by WAYBEL_0:34;
          then sup rngN <= sup downarrow v1 by A21,A26,YELLOW_0:34;
          hence contradiction by A11,A25,WAYBEL_0:34;
        end;
        then consider w be object such that
A27:    w in rngN and
A28:    not w in downarrow v1;
        reconsider w9 = w as Element of T by A27;
        consider i being object such that
A29:    i in dom the mapping of N and
A30:    w9 = (the mapping of N).i by A27,FUNCT_1:def 3;
        reconsider i as Element of N by A29;
        consider b being Element of N such that
A31:    for l being Element of N st b <= l holds N.i <= N.l by WAYBEL_0:11;
        take b;
        thus b in the carrier of N;
         reconsider QQ=Q as set by TARSKI:1;
        take QQ;
        thus QQ = Q;
        thus for j,k being Element of N st j = b & k >= j holds N.k in QQ
        proof
          let j,k be Element of N such that
A32:      j = b and
A33:      k >= j;
A34:      N.i <= N.k by A31,A32,A33;
          not N.i <= v1 by A28,A30,WAYBEL_0:17;
          then not N.k <= v1 by A34,ORDERS_2:3;
          then not N.k in downarrow v1 by WAYBEL_0:17;
          hence thesis by A24,XBOOLE_0:def 5;
        end;
      end;
      consider f be Function such that
A35:  dom f = Z & rng f c= the carrier of N and
A36:  for Q being object st Q in Z holds P[Q,f.Q] from FUNCT_1:sch 6(A22);
      reconsider rngf = rng f as finite Subset of [#]N by A19,A35,FINSET_1:8;
      [#]N is directed by WAYBEL_0:def 6;
      then consider k being Element of N such that
      k in [#]N and
A37:  k is_>=_than rngf by WAYBEL_0:1;
      take k;
      let k1 be Element of N such that
A38:  k <= k1;
      now
        let Q be set;
        assume
A39:    Q in Z;
        then
A40:    f.Q in rngf by A35,FUNCT_1:def 3;
        then reconsider j = f.Q as Element of N;
A41:     j <= k by A37,A40;
        P[Q,f.Q] by A36,A39;
        hence N.k1 in Q by A38,ORDERS_2:3,A41;
      end;
      then
A42:  N.k1 in Y1 by A20,SETFAM_1:43;
      Y1 c= union Y by A17,ZFMISC_1:74;
      then
A43:  N.k1 in Int V by A15,A42;
      Int V c= V by TOPS_1:16;
      hence thesis by A43;
    end;
  end;
  hence thesis;
end;
