
theorem Th20:
  for T be Lawson complete TopLattice for X be finite Subset of
  T holds (uparrow X)` is open & (downarrow X)` is open
proof
  let T be Lawson complete TopLattice;
  let X be finite Subset of T;
  deffunc F(Element of T) = uparrow $1;
  { uparrow x where x is Element of T : x in X } c= bool the carrier of T
  proof
    let z be object;
    assume z in { uparrow x where x is Element of T : x in X };
    then ex x be Element of T st z = uparrow x & x in X;
    hence thesis;
  end;
  then reconsider upx = { uparrow x where x is Element of T : x in X } as
  Subset-Family of T;
  { downarrow x where x is Element of T : x in X } c= bool the carrier of T
  proof
    let z be object;
    assume z in { downarrow x where x is Element of T : x in X };
    then ex x be Element of T st z = downarrow x & x in X;
    hence thesis;
  end;
  then reconsider dox = { downarrow x where x is Element of T : x in X } as
  Subset-Family of T;
  reconsider dox as Subset-Family of T;
  reconsider upx as Subset-Family of T;
A1: uparrow X = union upx by YELLOW_9:4;
  now
    let P be Subset of T;
    assume P in upx;
    then ex x be Element of T st P = uparrow x & x in X;
    hence P is closed by WAYBEL19:38;
  end;
  then
A2: upx is closed by TOPS_2:def 2;
A3: X is finite;
  {F(x) where x is Element of T : x in X } is finite from FRAENKEL:sch 21
  (A3);
  then uparrow X is closed by A2,A1,TOPS_2:21;
  hence (uparrow X)` is open by TOPS_1:3;
  deffunc F(Element of T) = downarrow $1;
A4: downarrow X = union dox by YELLOW_9:4;
  now
    let P be Subset of T;
    assume P in dox;
    then ex x be Element of T st P = downarrow x & x in X;
    hence P is closed by WAYBEL19:38;
  end;
  then
A5: dox is closed by TOPS_2:def 2;
  {F(x) where x is Element of T : x in X } is finite from FRAENKEL:sch 21
  (A3);
  then downarrow X is closed by A5,A4,TOPS_2:21;
  hence thesis by TOPS_1:3;
end;
