
theorem
  for N being Hausdorff topological_semilattice with_open_semilattices
  Lawson complete TopLattice holds N is with_compact_semilattices
proof
  let N be Hausdorff topological_semilattice with_open_semilattices Lawson
  complete TopLattice;
  let x be Point of N;
  consider BO being Basis of x such that
A1: for A being Subset of N st A in BO holds subrelstr A is
  meet-inheriting by Def4;
  set BC = {Cl A where A is Subset of N: A in BO};
  BC c= bool the carrier of N
  proof
    let k be object;
    assume k in BC;
    then ex A being Subset of N st k = Cl A & A in BO;
    hence thesis;
  end;
  then reconsider BC as Subset-Family of N;
  BC is basis of x
  proof
    let S be a_neighborhood of x;
A2: Int S c= S by TOPS_1:16;
    x in Int S by CONNSP_2:def 1;
    then consider W being Subset of N such that
A3: W in BO and
A4: W c= Int S by YELLOW_8:13;
A5: W is open by A3,YELLOW_8:12;
    x in W by A3,YELLOW_8:12;
    then x in Int W by A5,TOPS_1:23;
    then reconsider T = W as a_neighborhood of x by CONNSP_2:def 1;
    per cases;
    suppose
A6:   W <> [#]N;
      x in W by A3,YELLOW_8:12;
      then {x} c= W by ZFMISC_1:31;
      then consider G being Subset of N such that
A7:   G is open and
A8:   {x} c= G and
A9:   Cl G c= W by A5,A6,CONNSP_2:20;
      x in G by A8,ZFMISC_1:31;
      then consider K being Subset of N such that
A10:  K in BO and
A11:  K c= G by A7,YELLOW_8:13;
A12:  K is open by A10,YELLOW_8:12;
A13:  Int K c= Int Cl K by PRE_TOPC:18,TOPS_1:19;
      x in K by A10,YELLOW_8:12;
      then x in Int K by A12,TOPS_1:23;
      then reconsider KK = Cl K as a_neighborhood of x by A13,CONNSP_2:def 1;
      take KK;
      thus KK in BC by A10;
      Cl K c= Cl G by A11,PRE_TOPC:19;
      then Cl K c= W by A9;
      then KK c= Int S by A4;
      hence thesis by A2;
    end;
    suppose
A14:  W = [#]N;
      take T;
      Cl [#]N = [#]N by TOPS_1:2;
      hence T in BC by A3,A14;
      thus thesis by A4,A2;
    end;
  end;
  then reconsider BC as basis of x;
  take BC;
  let A be Subset of N;
  assume A in BC;
  then consider C being Subset of N such that
A15: A = Cl C and
A16: C in BO;
  subrelstr C is meet-inheriting by A1,A16;
  hence subrelstr A is meet-inheriting by A15,Th31;
  thus thesis by A15,COMPTS_1:8;
end;
