
theorem Th29:
  for L be distributive complete LATTICE st L opp is
meet-continuous for p be Element of L st p is completely-irreducible holds (the
  carrier of L) \ downarrow p is Open Filter of L
proof
  let L be distributive complete LATTICE;
  assume
A1: L opp is meet-continuous;
  let p be Element of L;
  assume
A2: p is completely-irreducible;
  then consider q be Element of L such that
A3: p < q and
A4: for s be Element of L st p < s holds q <= s and
  uparrow p = {p} \/ uparrow q by Th20;
  defpred P[Element of L] means $1 <= q & not $1 <= p;
  reconsider F = { t where t is Element of L : P[t]} as Subset of L from
  DOMAIN_1:sch 7;
  not q <= p by A3,ORDERS_2:6;
  then
A5: q in F;
A6: now
    let x,y be Element of L;
    assume that
A7: x in F and
A8: y in F;
A9: ex x1 be Element of L st x1 = x & x1 <= q & not x1 <= p by A7;
    take z = x "/\" y;
A10: z <= x by YELLOW_0:23;
A11: ex y1 be Element of L st y1 = y & y1 <= q & not y1 <= p by A8;
A12: not z <= p
    proof
A13:  now
        let d be Element of L;
        assume d >= y & d >= p;
        then d > p by A11,ORDERS_2:def 6;
        hence q <= d by A4;
      end;
      assume
A14:  z <= p;
A15:  q >= p by A3,ORDERS_2:def 6;
      x = x "/\" q by A9,YELLOW_0:25
        .= x "/\" (y "\/" p) by A11,A15,A13,YELLOW_0:22
        .= z "\/" (x "/\" p) by WAYBEL_1:def 3
        .= (x "\/" z) "/\" (z "\/" p) by WAYBEL_1:5
        .= x "/\" (p "\/" z) by A10,YELLOW_0:24
        .= x "/\" p by A14,YELLOW_0:24;
      hence contradiction by A9,YELLOW_0:25;
    end;
    z <= q by A9,A10,ORDERS_2:3;
    hence z in F by A12;
    thus z <= x by YELLOW_0:23;
    thus z <= y by YELLOW_0:23;
  end;
  p is irreducible by A2,Th23;
  then
A16: p is prime by WAYBEL_6:27;
  not Top L in Irr L by Th21;
  then p <> Top L by A2,Def4;
  then (downarrow p)` is Filter of L by A16,WAYBEL_6:26;
  then reconsider V = (the carrier of L) \ downarrow p as Filter of L by
SUBSET_1:def 4;
  reconsider F as non empty filtered Subset of L by A5,A6,WAYBEL_0:def 2;
  reconsider F1 = F as non empty directed Subset of L opp by YELLOW_7:27;
  now
    let x be Element of L;
    assume
A17: x in V;
    take y = inf F;
    thus y in V
    proof
      now
        let r be Element of L;
        assume r in {p} "\/" F;
        then r in {p "\/" v where v is Element of L : v in F} by YELLOW_4:15;
        then consider v be Element of L such that
A18:    r = p "\/" v and
A19:    v in F;
        ex v1 be Element of L st v = v1 & v1 <= q & not v1 <= p by A19;
        then
A20:    p <> r by A18,YELLOW_0:24;
        p <= r by A18,YELLOW_0:22;
        then p < r by A20,ORDERS_2:def 6;
        hence q <= r by A4;
      end;
      then
A21:  q is_<=_than {p} "\/" F by LATTICE3:def 8;
A22:  ex_inf_of {p~} "/\" F1,L by YELLOW_0:17;
      ex_inf_of F,L by YELLOW_0:17;
      then
A23:  inf F = sup F1 by YELLOW_7:13;
A24:  {p~} = {p} by LATTICE3:def 6;
      assume not y in V;
      then y in downarrow p by XBOOLE_0:def 5;
      then y <= p by WAYBEL_0:17;
      then p = p "\/" y by YELLOW_0:24
        .= p~ "/\" (inf F)~ by YELLOW_7:23
        .= p~ "/\" sup F1 by A23,LATTICE3:def 6
        .= sup ({p~} "/\" F1) by A1,WAYBEL_2:def 6
        .= "/\"({p~} "/\" F1,L) by A22,YELLOW_7:13
        .= "/\"({p} "\/" F,L) by A24,Th5;
      then q <= p by A21,YELLOW_0:33;
      hence contradiction by A3,ORDERS_2:6;
    end;
    then not y in downarrow p by XBOOLE_0:def 5;
    then
A25: not y <= p by WAYBEL_0:17;
    now
      let D be non empty directed Subset of L;
      assume
A26:  y <= sup D;
      D \ downarrow p is non empty
      proof
        assume D \ downarrow p is empty;
        then D c= downarrow p by XBOOLE_1:37;
        then sup D <= sup (downarrow p) by WAYBEL_7:1;
        then y <= sup (downarrow p) by A26,ORDERS_2:3;
        hence contradiction by A25,WAYBEL_0:34;
      end;
      then consider d be object such that
A27:  d in D \ downarrow p;
      reconsider d as Element of L by A27;
      take d;
      thus d in D by A27,XBOOLE_0:def 5;
      not d in downarrow p by A27,XBOOLE_0:def 5;
      then not d <= p by WAYBEL_0:17;
      then d "/\" q <= q & not d "/\" q <= p by A3,A4,Th28,YELLOW_0:23;
      then y is_<=_than F & d "/\" q in F by YELLOW_0:33;
      then d "/\" q <= d & y <= d "/\" q by LATTICE3:def 8,YELLOW_0:23;
      hence y <= d by ORDERS_2:3;
    end;
    then
A28: y << y by WAYBEL_3:def 1;
    not x in downarrow p by A17,XBOOLE_0:def 5;
    then not x <= p by WAYBEL_0:17;
    then x "/\" q <= q & not x "/\" q <= p by A3,A4,Th28,YELLOW_0:23;
    then y is_<=_than F & x "/\" q in F by YELLOW_0:33;
    then x "/\" q <= x & y <= x "/\" q by LATTICE3:def 8,YELLOW_0:23;
    then y <= x by ORDERS_2:3;
    hence y << x by A28,WAYBEL_3:2;
  end;
  hence thesis by WAYBEL_6:def 1;
end;
