
theorem Th7: :: PROPOSITION 4.5
  for L be LATTICE holds L is algebraic iff ( L is continuous & for
  x,y be Element of L st x << y ex k be Element of L st k in the carrier of
  CompactSublatt L & x <= k & k <= y )
proof
  let L be LATTICE;
  thus L is algebraic implies ( L is continuous & for x,y be Element of L st x
<< y ex k be Element of L st k in the carrier of CompactSublatt L & x <= k & k
  <= y )
  proof
    assume
A1: L is algebraic;
    then
A2: L is up-complete satisfying_axiom_K;
    now
      let x be Element of L;
A3:   compactbelow x is non empty directed by A1;
      then
A4:   ex s be object st s in compactbelow x by XBOOLE_0:def 1;
      compactbelow x c= waybelow x by Th6;
      then
A5:   ex_sup_of waybelow x,L by A2,A4,WAYBEL_0:75;
      ex_sup_of compactbelow x,L by A2,A3,WAYBEL_0:75;
      then sup compactbelow x <= sup waybelow x by A5,Th6,YELLOW_0:34;
      then
A6:   x <= sup waybelow x by A2;
      waybelow x is_<=_than x by WAYBEL_3:9;
      then sup waybelow x <= x by A5,YELLOW_0:def 9;
      hence x = sup waybelow x by A6,ORDERS_2:2;
    end;
    then
A7: L is satisfying_axiom_of_approximation by WAYBEL_3:def 5;
    for x be Element of L holds waybelow x is non empty directed
    proof
      let x be Element of L;
      compactbelow x c= waybelow x by Th6;
      hence thesis by A1;
    end;
    hence L is continuous by A2,A7,WAYBEL_3:def 6;
    let x,y be Element of L;
    reconsider D = compactbelow y as non empty directed Subset of L by A1;
    assume
A8: x << y;
    y = sup D by A2;
    then consider d being Element of L such that
A9: d in D and
A10: x <= d by A8,WAYBEL_3:def 1;
    take d;
    d is compact by A9,Th4;
    hence d in the carrier of CompactSublatt L by Def1;
    thus thesis by A9,A10,Th4;
  end;
  assume that
A11: L is continuous and
A12: for x,y be Element of L st x << y ex k be Element of L st k in the
  carrier of CompactSublatt L & x <= k & k <= y;
  now

    let x be Element of L;
A13: now
      let z be Element of L;
      thus z is_>=_than waybelow x implies z is_>=_than compactbelow x
      proof
        assume
A14:    z is_>=_than waybelow x;
        now
          let d be Element of L;
          assume
A15:      d in compactbelow x;
          then d is compact by Th4;
          then
A16:      d << d by WAYBEL_3:def 2;
          d <= x by A15,Th4;
          then d << x by A16,WAYBEL_3:2;
          then d in waybelow x by WAYBEL_3:7;
          hence d <= z by A14,LATTICE3:def 9;
        end;
        hence thesis by LATTICE3:def 9;
      end;
      thus z is_>=_than compactbelow x implies z is_>=_than waybelow x
      proof
        assume
A17:    z is_>=_than compactbelow x;
        now
          let d be Element of L;
          assume d in waybelow x;
          then d << x by WAYBEL_3:7;
          then consider k be Element of L such that
A18:      k in the carrier of CompactSublatt L and
A19:      d <= k and
A20:      k <= x by A12;
          k is compact by A18,Def1;
          then k in compactbelow x by A20;
          then k <= z by A17,LATTICE3:def 9;
          hence d <= z by A19,ORDERS_2:3;
        end;
        hence thesis by LATTICE3:def 9;
      end;
    end;
    x = sup waybelow x & ex_sup_of waybelow x,L by A11,WAYBEL_0:75
,WAYBEL_3:def 5;
    hence x = sup compactbelow x by A13,YELLOW_0:47;
  end;
  then
A21: L is satisfying_axiom_K;
  for x be Element of L holds compactbelow x is non empty directed
  proof
    let x be Element of L;
    now
      let Y be finite Subset of compactbelow x;
      compactbelow x c= waybelow x by Th6;
      then Y is finite Subset of waybelow x by XBOOLE_1:1;
      then consider b be Element of L such that
A22:  b in waybelow x and
A23:  b is_>=_than Y by A11,WAYBEL_0:1;
      b << x by A22,WAYBEL_3:7;
      then consider c be Element of L such that
A24:  c in the carrier of CompactSublatt L and
A25:  b <= c and
A26:  c <= x by A12;
      take c;
      c is compact by A24,Def1;
      hence c in compactbelow x by A26;
      thus c is_>=_than Y by A23,A25,YELLOW_0:4;
    end;
    hence thesis by WAYBEL_0:1;
  end;
  hence thesis by A11,A21;
end;
