
theorem Th32:
  for N being meet-continuous Lawson complete TopLattice for S
being Scott TopAugmentation of N holds (for x being Point of S ex J being Basis
  of x st for W being Subset of S st W in J holds W is Filter of S) iff N is
  with_open_semilattices
proof
  let N be meet-continuous Lawson complete TopLattice, S be Scott
  TopAugmentation of N;
A1: the RelStr of N = the RelStr of S by YELLOW_9:def 4;
  hereby
    assume
A2: for x being Point of S ex J being Basis of x st for W being Subset
    of S st W in J holds W is Filter of S;
    thus N is with_open_semilattices
    proof
      let x be Point of N;
      defpred P[set] means ex U1 being Filter of N, F being finite Subset of N
, W1 being Subset of N st $1 = W1 & U1 \ uparrow F = $1 & x in $1 & W1 is open;
      consider SF being Subset-Family of N such that
A3:   for W being Subset of N holds W in SF iff P[W] from SUBSET_1:
      sch 3;
      reconsider SF as Subset-Family of N;
A4:   now
        reconsider BL = {O\uparrow F where O, F is Subset of N: O in sigma N &
        F is finite} as Basis of N by WAYBEL19:32;
A5:     BL c= the topology of N by TOPS_2:64;
        reconsider y = x as Point of S by A1;
        let W be Subset of N such that
A6:     W is open and
A7:     x in W;
        consider By being Basis of y such that
A8:     for A being Subset of S st A in By holds A is Filter of S by A2;
        W = union { G where G is Subset of N: G in BL & G c= W } by A6,
YELLOW_8:9;
        then consider K being set such that
A9:     x in K and
A10:    K in { G where G is Subset of N: G in BL & G c= W } by A7,TARSKI:def 4;
        consider G being Subset of N such that
A11:    K = G and
A12:    G in BL and
A13:    G c= W by A10;
        consider V, F being Subset of N such that
A14:    G = V \ uparrow F and
A15:    V in sigma N and
A16:    F is finite by A12;
        reconsider F as finite Subset of N by A16;
A17:    not x in uparrow F by A9,A11,A14,XBOOLE_0:def 5;
        reconsider V as Subset of S by A1;
A18:    y in V by A9,A11,A14,XBOOLE_0:def 5;
A19:    sigma N = sigma S by A1,YELLOW_9:52;
        then V is open by A15,WAYBEL14:24;
        then consider U1 being Subset of S such that
A20:    U1 in By and
A21:    U1 c= V by A18,YELLOW_8:13;
        reconsider U2 = U1 as Subset of N by A1;
        U1 is Filter of S by A8,A20;
        then reconsider U2 as Filter of N by A1,WAYBEL_0:4,25;
        U2 \ uparrow F is Subset of N;
        then reconsider IT = U1 \ uparrow F as Subset of N;
        take U2, F, IT;
        thus IT = U2 \ uparrow F;
        y in U1 by A20,YELLOW_8:12;
        hence x in IT by A17,XBOOLE_0:def 5;
        U1 is open by A20,YELLOW_8:12;
        then U1 in sigma S by WAYBEL14:24;
        then IT in BL by A19;
        hence IT is open by A5;
        IT c= G by A14,A21,XBOOLE_1:33;
        hence IT c= W by A13;
      end;
      SF is Basis of x
      proof
A22:    SF is open
        proof
          let a be Subset of N;
          assume
A23:      a in SF;
          reconsider W = a as Subset of N;
          ex U1 being Filter of N, F being finite Subset of
N, W1 being Subset of N st W1 = W & U1 \ uparrow F = W & x in W & W1 is
          open by A3,A23;
          hence thesis;
        end;
        SF is x-quasi_basis
        proof
        for a being set st a in SF holds x in a
        proof
          let a be set;
          assume
A24:      a in SF;
          then reconsider W = a as Subset of N;
          ex U1 being Filter of N, F being finite Subset of
N, W1 being Subset of N st W1 = W & U1 \ uparrow F = W & x in W & W1 is
          open by A3,A24;
          hence thesis;
        end;
        hence x in Intersect SF by SETFAM_1:43;
        let W be Subset of N;
        assume that
A25:    W is open and
A26:    x in W;
        consider U2 being Filter of N, F being finite Subset of N, IT being
        Subset of N such that
A27:    IT = U2 \ uparrow F and
A28:    x in IT and
A29:    IT is open and
A30:    IT c= W by A25,A26,A4;
        take IT;
        thus thesis by A3,A27,A28,A29,A30;
      end;
      hence thesis by A22;
      end;
      then reconsider SF as Basis of x;
      take SF;
      let W be Subset of N;
      assume W in SF;
      then consider U1 being Filter of N, F being finite Subset of N, W1 being
      Subset of N such that
      W1 = W and
A31:  U1 \ uparrow F = W and
      x in W and
      W1 is open by A3;
      set SW = subrelstr W;
      thus SW is meet-inheriting
      proof
        let a, b be Element of N such that
A32:    a in the carrier of SW and
A33:    b in the carrier of SW and
        ex_inf_of {a,b},N;
A34:    the carrier of SW = W by YELLOW_0:def 15;
        then
A35:    b in U1 by A31,A33,XBOOLE_0:def 5;
A36:    not a in uparrow F by A34,A31,A32,XBOOLE_0:def 5;
        for y being Element of N st y <= a "/\" b holds not y in F
        proof
A37:      a "/\" b <= a by YELLOW_0:23;
          let y be Element of N;
          assume y <= a "/\" b;
          then y <= a by A37,ORDERS_2:3;
          hence thesis by A36,WAYBEL_0:def 16;
        end;
        then
A38:    not a "/\" b in uparrow F by WAYBEL_0:def 16;
        a in U1 by A34,A31,A32,XBOOLE_0:def 5;
        then consider z being Element of N such that
A39:    z in U1 and
A40:    z <= a and
A41:    z <= b by A35,WAYBEL_0:def 2;
        z "/\" z <= a "/\" b by A40,A41,YELLOW_3:2;
        then z <= a "/\" b by YELLOW_0:25;
        then a "/\" b in U1 by A39,WAYBEL_0:def 20;
        then a "/\" b in W by A38,A31,XBOOLE_0:def 5;
        hence thesis by A34,YELLOW_0:40;
      end;
    end;
  end;
  assume
A42: N is with_open_semilattices;
  let x be Point of S;
  reconsider y = x as Point of N by A1;
  consider J being Basis of y such that
A43: for A being Subset of N st A in J holds subrelstr A is
  meet-inheriting by A42;
  reconsider J9 = {uparrow A where A is Subset of N: A in J} as Basis of x by
Th16;
  take J9;
  let W be Subset of S;
  assume W in J9;
  then consider V being Subset of N such that
A44: W = uparrow V and
A45: V in J;
  subrelstr V is meet-inheriting by A43,A45;
  then
A46: V is filtered by YELLOW12:26;
  x in V by A45,YELLOW_8:12;
  hence thesis by A46,A1,A44,WAYBEL_0:4,25;
end;
