
theorem Th30:
  for N being Lawson complete TopLattice holds N is continuous
  iff N is meet-continuous Hausdorff
proof
  let N be Lawson complete TopLattice;
  thus N is continuous implies N is meet-continuous Hausdorff;
  assume
A1: N is meet-continuous Hausdorff;
  thus
A2: for x being Element of N holds waybelow x is non empty directed;
  thus N is up-complete;
  for x, y being Element of N st not x <= y ex u being Element of N st u
  << x & not u <= y
  proof
    reconsider J = {O\uparrow F where O, F is Subset of N: O in sigma N & F is
    finite} as Basis of N by WAYBEL19:32;
    set S = the Scott TopAugmentation of N;
A3: for N being Semilattice, x, y being Element of N st ex u being
Element of N st u << x & not u <= x "/\" y ex u being Element of N st u << x &
    not u <= y
    proof
      let N be Semilattice, x, y be Element of N;
      given u being Element of N such that
A4:   u << x and
A5:   not u <= x "/\" y;
      take u;
      thus u << x by A4;
      assume
A6:   u <= y;
      u <= x by A4,WAYBEL_3:1;
      then u "/\" u <= x "/\" y by A6,YELLOW_3:2;
      hence contradiction by A5,YELLOW_0:25;
    end;
    let x, y be Element of N;
    assume not x <= y;
    then x "/\" y <> x by YELLOW_0:23;
    then consider W, V being Subset of N such that
A7: W is open and
A8: V is open and
A9: x in W and
A10: x "/\" y in V and
A11: W misses V by A1;
    V = union {G where G is Subset of N: G in J & G c= V} by A8,YELLOW_8:9;
    then consider K being set such that
A12: x "/\" y in K and
A13: K in {G where G is Subset of N: G in J & G c= V} by A10,TARSKI:def 4;
    consider V1 being Subset of N such that
A14: K = V1 and
A15: V1 in J and
A16: V1 c= V by A13;
    consider U2, F being Subset of N such that
A17: V1 = U2\uparrow F and
A18: U2 in sigma N and
A19: F is finite by A15;
A20: the RelStr of S = the RelStr of N by YELLOW_9:def 4;
    then reconsider x1 = x, y1 = y as Element of S;
A21: ex_inf_of {x1,y1},S by YELLOW_0:21;
    reconsider U1 = U2 as Subset of S by A20;
    reconsider WU1 = W /\ U2 as Subset of N;
    reconsider W1 = WU1 as Subset of S by A20;
A22: uparrow W1 = uparrow WU1 by A20,WAYBEL_0:13;
    U2 in sigma S by A20,A18,YELLOW_9:52;
    then
A23: U1 is open by WAYBEL14:24;
    then
A24: U1 is upper by WAYBEL11:def 4;
    WU1 c= uparrow F
    proof
A25:  W misses V1 by A11,A16,XBOOLE_1:63;
A26:  WU1 \ uparrow F c= U1 \ uparrow F by XBOOLE_1:17,33;
      let w be object;
      assume that
A27:  w in WU1 and
A28:  not w in uparrow F;
A29:  w in W by A27,XBOOLE_0:def 4;
      w in WU1 \ uparrow F by A27,A28,XBOOLE_0:def 5;
      hence contradiction by A26,A17,A29,A25,XBOOLE_0:3;
    end;
    then WU1 is_coarser_than uparrow F by WAYBEL12:16;
    then
A30: uparrow WU1 c= uparrow F by YELLOW12:28;
A31: x1 "/\" y1 <= x1 by YELLOW_0:23;
    x "/\" y = inf {x,y} by YELLOW_0:40
      .= "/\"({x1,y1},S) by A20,A21,YELLOW_0:27
      .= x1 "/\" y1 by YELLOW_0:40;
    then x1 "/\" y1 in U1 by A12,A14,A17,XBOOLE_0:def 5;
    then x1 in U1 by A24,A31;
    then
A32: x in W1 by A9,XBOOLE_0:def 4;
    W1 c= uparrow W1 by WAYBEL_0:16;
    then
A33: x in uparrow W1 by A32;
    reconsider F1 = F as finite Subset of S by A20,A19;
A34: uparrow F1 = uparrow F by A20,WAYBEL_0:13;
    N is Lawson correct TopAugmentation of S by A20,YELLOW_9:def 4;
    then U2 is open by A23,WAYBEL19:37;
    then uparrow W1 c= Int uparrow F1 by A7,A1,A30,A22,A34,Th15,TOPS_1:24;
    then
A35: x in Int uparrow F1 by A33;
    S is meet-continuous by A1,A20,YELLOW12:14;
    then Int uparrow F1 c= union { wayabove u where u is Element of S : u in
    F1 } by Th29;
    then consider A being set such that
A36: x in A and
A37: A in { wayabove u where u is Element of S : u in F1 } by A35,TARSKI:def 4;
    consider u being Element of S such that
A38: A = wayabove u and
A39: u in F1 by A37;
    reconsider u1 = u as Element of N by A20;
A40: wayabove u1 = wayabove u by A20,YELLOW12:13;
    now
      take u1;
      thus u1 << x by A36,A38,A40,WAYBEL_3:8;
      uparrow u1 c= uparrow F by A39,YELLOW12:30;
      then not x "/\" y in uparrow u1 by A12,A14,A17,XBOOLE_0:def 5;
      hence not u1 <= x "/\" y by WAYBEL_0:18;
    end;
    then consider u2 being Element of N such that
A41: u2 << x and
A42: not u2 <= y by A3;
    take u2;
    thus thesis by A41,A42;
  end;
  hence thesis by A2,WAYBEL_3:24;
end;
