
theorem Th24:
  for L being complete LATTICE for S being closure System of L
  holds closure_op S is directed-sups-preserving iff S is
  directed-sups-inheriting
proof
  let L be complete LATTICE;
  let S be closure System of L;
  set c = closure_op S;
A1: Image c = the RelStr of S by Th18;
  hereby
    set Lk = {k where k is Element of L: c.k <= k};
    set k = the Element of L;
A2: Lk c= the carrier of L
    proof
      let x be object;
      assume x in Lk;
      then ex k being Element of L st x = k & c.k <= k;
      hence thesis;
    end;
    c.(c.k) = c.k by YELLOW_2:18;
    then
A3: c.k in Lk;
    assume closure_op S is directed-sups-preserving;
    then
A4: Image c is directed-sups-inheriting by A3,A2,WAYBEL_1:52;
    thus S is directed-sups-inheriting
    proof
      let X be directed Subset of S such that
A5:   X <> {} and
A6:   ex_sup_of X,L;
      reconsider Y = X as Subset of Image c by A1;
      Y is directed by A1,WAYBEL_0:3;
      hence thesis by A1,A4,A5,A6;
    end;
  end;
  assume
A7: for X being directed Subset of S st X <> {} & ex_sup_of X,L holds
  "\/"(X,L) in the carrier of S;
  let X be Subset of L such that
A8: X is non empty directed;
  rng c = the carrier of S by A1,YELLOW_0:def 15;
  then reconsider Y = c.:X as Subset of S by RELAT_1:111;
  assume ex_sup_of X,L;
  thus ex_sup_of c.:X,L by YELLOW_0:17;
  c.:X is_<=_than c.sup X
  proof
    let x be Element of L;
    assume x in c.:X;
    then consider a being object such that
A9: a in the carrier of L and
A10: a in X and
A11: x = c.a by FUNCT_2:64;
    reconsider a as Element of L by A9;
    a <= sup X by A10,YELLOW_2:22;
    hence thesis by A11,WAYBEL_1:def 2;
  end;
  then
A12: sup (c.:X) <= c.sup X by YELLOW_0:32;
  X is_<=_than sup (c.:X)
  proof
    let x be Element of L;
    assume x in X;
    then c.x in c.:X by FUNCT_2:35;
    then
A13: c.x <= sup (c.:X) by YELLOW_2:22;
    x <= c.x by Th5;
    hence thesis by A13,ORDERS_2:3;
  end;
  then
A14: sup X <= sup (c.:X) by YELLOW_0:32;
  set x = the Element of X;
  x in X by A8;
  then
A15: c.x in c.:X by FUNCT_2:35;
  Y is directed by A8,Th23,YELLOW_2:15;
  then sup (c.:X) in the carrier of S by A7,A15,YELLOW_0:17;
  then ex a being Element of L st c.a = sup (c.:X) by A1,YELLOW_2:10;
  then c.sup (c.:X) = sup (c.:X) by YELLOW_2:18;
  then c.sup X <= sup (c.:X) by A14,WAYBEL_1:def 2;
  hence thesis by A12,ORDERS_2:2;
end;
