
theorem Th23:
  for L be algebraic LATTICE for c be closure Function of L,L st c
is directed-sups-preserving holds c.:([#]CompactSublatt L) c= [#]CompactSublatt
  Image c
proof
  let L be algebraic LATTICE;
  let c be closure Function of L,L;
  assume
A1: c is directed-sups-preserving;
  let x be object;
A2: c is idempotent monotone by WAYBEL_1:def 13;
  assume x in c.:([#]CompactSublatt L);
  then consider y be object such that
A3: y in dom c and
A4: y in [#]CompactSublatt L and
A5: x = c.y by FUNCT_1:def 6;
A6: x in rng c by A3,A5,FUNCT_1:def 3;
  then reconsider x9 = x as Element of Image c by YELLOW_0:def 15;
  reconsider x1 = x9 as Element of L by A6;
  reconsider y9 = y as Element of L by A3;
  y9 is compact by A4,Def1;
  then
A7: y9 << y9 by WAYBEL_3:def 2;
  now
    id(L) <= c by WAYBEL_1:def 14;
    then id(L).y9 <= c.y9 by YELLOW_2:9;
    then
A8: y9 <= x1 by A5,FUNCT_1:18;
    let D be non empty directed Subset of Image c;
    assume
A9: x9 <= sup D;
    D c= the carrier of Image c;
    then
A10: D c= rng c by YELLOW_0:def 15;
    then reconsider D9 = D as non empty (Subset of L) by XBOOLE_1:1;
    reconsider D9 as non empty directed (Subset of L) by YELLOW_2:7;
A11: ex_sup_of D9,L by WAYBEL_0:75;
    c preserves_sup_of D9 by A1,WAYBEL_0:def 37;
    then
A12: c.sup D9 = sup (c.:D9) by A11,WAYBEL_0:def 31
      .= sup D9 by A2,A10,YELLOW_2:20;
    c.sup D9 = sup D by A11,WAYBEL_1:55;
    then x1 <= sup D9 by A9,A12,YELLOW_0:59;
    then y9 <= sup D9 by A8,ORDERS_2:3;
    then consider d9 be Element of L such that
A13: d9 in D9 and
A14: y9 <= d9 by A7,WAYBEL_3:def 1;
    reconsider d = d9 as Element of Image c by A13;
    take d;
    thus d in D by A13;
    d in the carrier of Image c;
    then d in rng c by YELLOW_0:def 15;
    then d9 in {z where z is Element of L: z = c.z} by A2,YELLOW_2:19;
    then
A15: ex z9 be Element of L st d9 = z9 & z9 = c.z9;
    c.y9 <= c.d9 by A2,A14,WAYBEL_1:def 2;
    hence x9 <= d by A5,A15,YELLOW_0:60;
  end;
  then x9 << x9 by WAYBEL_3:def 1;
  then x9 is compact by WAYBEL_3:def 2;
  hence thesis by Def1;
end;
