reserve a for set;
reserve L for lower-bounded sup-Semilattice;
reserve x for Element of L;

theorem Th38:
  for L being lower-bounded continuous LATTICE holds L is meet-continuous
proof
  let L be lower-bounded continuous LATTICE;
  now
    let D be non empty directed Subset of L;
    let x be Element of L;
    assume
A1: x <= sup D;
A2: ex_sup_of waybelow x,L by YELLOW_0:17;
A3: ex_sup_of downarrow (sup ({x} "/\" D)),L by YELLOW_0:17;
    waybelow x c= downarrow (sup ({x} "/\" D))
    proof
      let q be object;
      assume q in waybelow x;
      then q in {y where y is Element of L: y << x} by WAYBEL_3:def 3;
      then consider q9 be Element of L such that
A4:   q9 = q and
A5:   q9 << x;
A6:   q9 <= x by A5,WAYBEL_3:1;
      consider d be Element of L such that
A7:   d in D and
A8:   q9 <= d by A1,A5,WAYBEL_3:def 1;
      x in {x} by TARSKI:def 1;
      then x "/\" d in { a "/\" b where a, b is Element of L : a in {x} & b in
      D } by A7;
      then
A9:   x "/\" d in {x} "/\" D by YELLOW_4:def 4;
      ex_inf_of {x,d},L by YELLOW_0:17;
      then
A10:  q9 <= x "/\" d by A6,A8,YELLOW_0:19;
      sup ({x} "/\" D) is_>=_than {x} "/\" D by YELLOW_0:32;
      then x "/\" d <= sup ({x} "/\" D) by A9;
      then q9 <= sup ({x} "/\" D) by A10,ORDERS_2:3;
      hence thesis by A4,WAYBEL_0:17;
    end;
    then sup waybelow x <= sup downarrow (sup ({x} "/\" D)) by A2,A3,
YELLOW_0:34;
    then sup waybelow x <= sup ({x} "/\" D) by WAYBEL_0:34;
    then
A11: x <= sup ({x} "/\" D) by WAYBEL_3:def 5;
    sup ({x} "/\" D) <= x by Lm11;
    hence x = sup ({x} "/\" D) by A11,ORDERS_2:2;
  end;
  then for x being Element of L, D being non empty directed Subset of L
  st x <= sup D holds x = sup ({x} "/\" D);
  hence thesis by WAYBEL_2:52;
end;
