
theorem

:: 1.7. LEMMA, p. 145
  for T being Lawson complete TopLattice for F being non empty
  filtered Subset of T holds Lim (F opp+id) = {inf F}
proof
  let T be Lawson complete TopLattice;
  reconsider K = (sigma T) \/ the set of all
(uparrow x)` where x is Element of T as prebasis of T by Th30;
  set S = the Scott TopAugmentation of T;
  let F be non empty filtered Subset of T;
  set N = F opp+id;
A1: the carrier of N = F by YELLOW_9:7;
A2: N is full SubRelStr of T opp by YELLOW_9:7;
  thus Lim N c= {inf F}
  proof
    let p be object;
    assume
A3: p in Lim N;
    then reconsider p as Element of T;
    the mapping of N = id F by Th27;
    then rng the mapping of N = F by RELAT_1:45;
    then
A4: p in Cl F by A3,WAYBEL_9:24;
    p is_<=_than F
    proof
      let u be Element of T;
      assume
A5:   u in F;
A6:   N is_eventually_in downarrow u
      proof
        reconsider i = u as Element of N by A5,YELLOW_9:7;
        take i;
        let j be Element of N;
        j in F by A1;
        then reconsider z = j as Element of T;
        assume j >= i;
        then z~ >= u~ by A2,YELLOW_0:59;
        then z <= u by LATTICE3:9;
        then z in downarrow u by WAYBEL_0:17;
        hence thesis by YELLOW_9:7;
      end;
      downarrow u is closed by Th38;
      then Cl downarrow u = downarrow u by PRE_TOPC:22;
      then Lim N c= downarrow u by A6,Th26;
      hence thesis by A3,WAYBEL_0:17;
    end;
    then
A7: p <= inf F by YELLOW_0:33;
    inf F is_<=_than F by YELLOW_0:33;
    then
A8: F c= uparrow inf F by YELLOW_2:2;
    uparrow inf F is closed by Th38;
    then Cl uparrow inf F = uparrow inf F by PRE_TOPC:22;
    then Cl F c= uparrow inf F by A8,PRE_TOPC:19;
    then p >= inf F by A4,WAYBEL_0:18;
    then p = inf F by A7,ORDERS_2:2;
    hence thesis by TARSKI:def 1;
  end;
A9: the topology of S = sigma T by YELLOW_9:51;
A10: the RelStr of S = the RelStr of T by YELLOW_9:def 4;
  now
    let A be Subset of T;
    assume that
A11: inf F in A and
A12: A in K;
    per cases by A12,XBOOLE_0:def 3;
    suppose
A13:  A in sigma T;
      then reconsider a = A as Subset of S by A9;
      set i = the Element of N;
      a is open by A9,A13;
      then a is upper by WAYBEL11:def 4;
      then
A14:  A is upper by A10,WAYBEL_0:25;
      thus N is_eventually_in A
      proof
        take i;
        let j be Element of N;
        j in F by A1;
        then reconsider z = j as Element of T;
        z >= inf F by A1,YELLOW_2:22;
        then z in A by A11,A14;
        hence thesis by YELLOW_9:7;
      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
A15:  A = (uparrow x)`;
      not inf F >= x by A11,A15,YELLOW_9:1;
      then not F is_>=_than x by YELLOW_0:33;
      then consider y being Element of T such that
A16:  y in F and
A17:  not y >= x;
      thus N is_eventually_in A
      proof
        reconsider i = y as Element of N by A16,YELLOW_9:7;
        take i;
        let j be Element of N;
        j in F by A1;
        then reconsider z = j as Element of T;
        assume j >= i;
        then z~ >= y~ by A2,YELLOW_0:59;
        then z <= y by LATTICE3:9;
        then not z >= x by A17,ORDERS_2:3;
        then z in A by A15,YELLOW_9:1;
        hence thesis by YELLOW_9:7;
      end;
    end;
  end;
  then inf F in Lim N by Th25;
  hence thesis by ZFMISC_1:31;
end;
