
theorem Th44:

:: 1.7. LEMMA, -- WAYBEL19:44 revised
  for T being Lawson complete TopLattice
  for N being eventually-filtered net of T holds Lim N = {inf N}
proof
  let T be Lawson complete TopLattice;
  set S = the Scott TopAugmentation of T;
  let N be eventually-filtered net of T;
  reconsider F = rng the mapping of N
  as filtered non empty Subset of T by Th43;
A1: the topology of S = sigma T by YELLOW_9:51;
A2: the RelStr of S = the RelStr of T by YELLOW_9:def 4;
A3: inf N = Inf the mapping of N by WAYBEL_9:def 2
    .= "/\"(F, T) by YELLOW_2:def 6;
A4: dom the mapping of N = the carrier of N by FUNCT_2:def 1;
  thus Lim N c= {inf N}
  proof
    let p be object;
    assume
A5: p in Lim N;
    then reconsider p as Element of T;
    p is_<=_than F
    proof
      let u be Element of T;
      assume u in F;
      then consider i being object such that
A6:   i in dom the mapping of N and
A7:   u = (the mapping of N).i by FUNCT_1:def 3;
      reconsider i as Element of N by A6;
      consider ii being Element of N such that
A8:   for k being Element of N st ii <= k holds N.i >= N.k by WAYBEL_0:12;
      downarrow u is closed by WAYBEL19:38;
      then
A9:   Cl downarrow u = downarrow u by PRE_TOPC:22;
      N is_eventually_in downarrow u
      proof
        take ii;
        let j be Element of N;
        assume j >= ii;
        then N.j <= N.i by A8;
        hence thesis by A7,WAYBEL_0:17;
      end;
      then Lim N c= downarrow u by A9,WAYBEL19:26;
      hence thesis by A5,WAYBEL_0:17;
    end;
    then
A10: p <= inf N by A3,YELLOW_0:33;
    inf N is_<=_than F by A3,YELLOW_0:33;
    then
A11: F c= uparrow inf N by YELLOW_2:2;
    uparrow inf N is closed by WAYBEL19:38;
    then Cl uparrow inf N = uparrow inf N by PRE_TOPC:22;
    then
A12: Cl F c= uparrow inf N by A11,PRE_TOPC:19;
    p in Cl F by A5,WAYBEL_9:24;
    then p >= inf N by A12,WAYBEL_0:18;
    then p = inf N by A10,ORDERS_2:2;
    hence thesis by TARSKI:def 1;
  end;
  reconsider K = (sigma T) \/
  the set of all (uparrow x)` where x is Element of T as prebasis of T
  by WAYBEL19:30;
  now
    let A be Subset of T;
    assume that
A13: inf F in A and
A14: A in K;
    per cases by A14,XBOOLE_0:def 3;
    suppose
A15:  A in sigma T;
      then reconsider a = A as Subset of S by A1;
      a is open by A1,A15,PRE_TOPC:def 2;
      then a is upper by WAYBEL11:def 4;
      then
A16:  A is upper by A2,WAYBEL_0:25;
      set i = the Element of N;
      thus N is_eventually_in A
      proof
        take i;
        let j be Element of N;
        N.j in F by A4,FUNCT_1:def 3;
        then N.j >= inf F by YELLOW_2:22;
        hence thesis by A13,A16;
      end;
    end;
    suppose
      A in the set of all (uparrow x)` where x is Element of T;
      then consider x being Element of T such that
A17:  A = (uparrow x)`;
      not inf F >= x by A13,A17,YELLOW_9:1;
      then not F is_>=_than x by YELLOW_0:33;
      then consider y being Element of T such that
A18:  y in F and
A19:  not y >= x;
      consider i being object such that
A20:  i in the carrier of N and
A21:  y = (the mapping of N).i by A4,A18,FUNCT_1:def 3;
      thus N is_eventually_in A
      proof
        reconsider i as Element of N by A20;
        consider ii being Element of N such that
A22:    for k being Element of N st ii <= k holds N.i >= N.k by WAYBEL_0:12;
        take ii;
        let j be Element of N;
        assume j >= ii;
        then N.j <= N.i by A22;
        then not N.j >= x by A19,A21,ORDERS_2:3;
        hence thesis by A17,YELLOW_9:1;
      end;
    end;
  end;
  then inf F in Lim N by WAYBEL19:25;
  hence thesis by A3,ZFMISC_1:31;
end;
