
theorem :: PROPOSITION 4.9.(ii)
  for L be algebraic lower-bounded LATTICE, c be closure Function of L,L
  st c is directed-sups-preserving holds c.:([#]CompactSublatt L) = [#]
  CompactSublatt Image c
proof
  let L be algebraic lower-bounded LATTICE;
  let c be closure Function of L,L;
  assume
A1: c is directed-sups-preserving;
  now
    c is idempotent by WAYBEL_1:def 13;
    then
A2: rng c = {x where x is Element of L: x = c.x} by YELLOW_2:19;
    c is idempotent by WAYBEL_1:def 13;
    then reconsider Imc = Image c as complete LATTICE by A1,YELLOW_2:35;
    let a9 be object;
    assume
A3: a9 in [#]CompactSublatt Image c;
A4: the carrier of CompactSublatt Imc c= the carrier of Imc by YELLOW_0:def 13;
    then reconsider a = a9 as Element of Imc by A3;
    a is compact by A3,Def1;
    then
A5: a << a by WAYBEL_3:def 2;
    a9 in the carrier of Imc by A3,A4;
    then a in rng c by YELLOW_0:def 15;
    then consider a1 be Element of L such that
A6: a = a1 and
A7: a1 = c.a1 by A2;
    compactbelow a1 is non empty directed Subset of L by Def4;
    then
A8: c preserves_sup_of compactbelow a1 & ex_sup_of compactbelow a1,L by A1,
WAYBEL_0:75,def 37;
A9: c is monotone by A1,YELLOW_2:16;
    now
      let z be object;
      assume z in c.:(compactbelow a1);
      then consider v be object such that
A10:  v in dom c and
A11:  v in compactbelow a1 and
A12:  z = c.v by FUNCT_1:def 6;
      reconsider v9 = v as Element of L by A11;
A13:  v in downarrow a1 /\ the carrier of CompactSublatt L by A11,Th5;
      then v in downarrow a1 by XBOOLE_0:def 4;
      then v9 <= a1 by WAYBEL_0:17;
      then c.v9 <= a1 by A7,A9,WAYBEL_1:def 2;
      then
A14:  z in (downarrow a1) by A12,WAYBEL_0:17;
      v in [#]CompactSublatt L by A13,XBOOLE_0:def 4;
      then z in c.:([#]CompactSublatt L) by A10,A12,FUNCT_1:def 6;
      hence z in (downarrow a1) /\ (c.:([#]CompactSublatt L)) by A14,
XBOOLE_0:def 4;
    end;
    then
A15: c.:(compactbelow a1) c= (downarrow a1) /\ (c.:([#] CompactSublatt L));
    a = sup compactbelow a1 by A6,Def3;
    then
A16: a = sup (c.:(compactbelow a1)) by A6,A7,A8,WAYBEL_0:def 31;
    c.:(compactbelow a1) c= rng c by RELAT_1:111;
    then
A17: c.:(compactbelow a1) c= the carrier of Imc by YELLOW_0:def 15;
A18: downarrow a /\ c.:([#]CompactSublatt L) is non empty directed Subset
    of Imc
    proof
      (c.:[#]CompactSublatt L) /\ downarrow a is Subset of Imc;
      then reconsider
      D = downarrow a /\ c.:[#]CompactSublatt L as Subset of Imc;
A19:  Bottom Imc <= a by YELLOW_0:44;
A20:  now
        let x,y be Element of Imc;
        assume that
A21:    x in D and
A22:    y in D;
        x in c.:([#]CompactSublatt L) by A21,XBOOLE_0:def 4;
        then consider d be object such that
A23:    d in dom c and
A24:    d in [#]CompactSublatt L and
A25:    x = c.d by FUNCT_1:def 6;
        y in c.: ([#]CompactSublatt L) by A22,XBOOLE_0:def 4;
        then consider e be object such that
A26:    e in dom c and
A27:    e in [#]CompactSublatt L and
A28:    y = c.e by FUNCT_1:def 6;
        reconsider e as Element of L by A26;
        y in downarrow a by A22,XBOOLE_0:def 4;
        then y <= a by WAYBEL_0:17;
        then
A29:    c.e <= a1 by A6,A28,YELLOW_0:59;
        reconsider d as Element of L by A23;
A30:    d <= d "\/" e by YELLOW_0:22;
        x in downarrow a by A21,XBOOLE_0:def 4;
        then x <= a by WAYBEL_0:17;
        then c.d <= a1 by A6,A25,YELLOW_0:59;
        then
A31:    c.d "\/" c.e <= a1 by A29,YELLOW_0:22;
        d "\/" e in the carrier of L;
        then d "\/" e in dom c by FUNCT_2:def 1;
        then c.(d "\/" e) in rng c by FUNCT_1:def 3;
        then reconsider z = c.(d "\/" e) as Element of Imc by YELLOW_0:def 15;
        take z;
A32:    id(L) <= c by WAYBEL_1:def 14;
        then id(L).e <= c.e by YELLOW_2:9;
        then
A33:    e <= c.e by FUNCT_1:18;
        id(L).d <= c.d by A32,YELLOW_2:9;
        then d <= c.d by FUNCT_1:18;
        then d "\/" e <= c.d "\/" c.e by A33,YELLOW_3:3;
        then d "\/" e <= a1 by A31,ORDERS_2:3;
        then c.(d "\/" e) <= a1 by A7,A9,WAYBEL_1:def 2;
        then z <= a by A6,YELLOW_0:60;
        then
A34:    z in downarrow a by WAYBEL_0:17;
A35:    e <= d "\/" e by YELLOW_0:22;
        then
A36:    c.e <= c.(d "\/" e) by A9,WAYBEL_1:def 2;
        e is compact by A27,Def1;
        then e << e by WAYBEL_3:def 2;
        then
A37:    e << d "\/" e by A35,WAYBEL_3:2;
        d is compact by A24,Def1;
        then d << d by WAYBEL_3:def 2;
        then d << d "\/" e by A30,WAYBEL_3:2;
        then d "\/" e << d "\/" e by A37,WAYBEL_3:3;
        then d "\/" e is compact by WAYBEL_3:def 2;
        then
A38:    d "\/" e in [#]CompactSublatt L by Def1;
        d "\/" e in the carrier of L;
        then d "\/" e in dom c by FUNCT_2:def 1;
        then z in c.:([#]CompactSublatt L) by A38,FUNCT_1:def 6;
        hence z in D by A34,XBOOLE_0:def 4;
        c.d <= c.(d "\/" e) by A9,A30,WAYBEL_1:def 2;
        hence x <= z & y <= z by A25,A28,A36,YELLOW_0:60;
      end;
      Bottom L is compact by WAYBEL_3:15;
      then dom c = the carrier of L & Bottom L in [#]CompactSublatt L by Def1,
FUNCT_2:def 1;
      then
A39:  c.Bottom L in c.:([#]CompactSublatt L) by FUNCT_1:def 6;
A40:  ex_sup_of {},L & {} c= the carrier of L by YELLOW_0:42;
A41:  {} c= the carrier of Imc;
      c.Bottom L = c."\/"({},L) by YELLOW_0:def 11
        .= "\/"({},Imc) by A40,A41,WAYBEL_1:55
        .= Bottom Imc by YELLOW_0:def 11;
      then c.Bottom L in downarrow a by A19,WAYBEL_0:17;
      hence thesis by A39,A20,WAYBEL_0:def 1,XBOOLE_0:def 4;
    end;
A42: c.:([#]CompactSublatt L) c= rng c by RELAT_1:111;
    now
      let z be object;
      assume
A43:  z in (downarrow a1) /\ (c.:([#]CompactSublatt L));
      then reconsider z1 = z as Element of L;
A44:  z in c.:([#] CompactSublatt L) by A43,XBOOLE_0:def 4;
      then reconsider z2 = z1 as Element of Imc by A42,YELLOW_0:def 15;
      z in downarrow a1 by A43,XBOOLE_0:def 4;
      then z1 <= a1 by WAYBEL_0:17;
      then z2 <= a by A6,YELLOW_0:60;
      then z in (downarrow a) by WAYBEL_0:17;
      hence z in (downarrow a) /\ (c.: ([#]CompactSublatt L)) by A44,
XBOOLE_0:def 4;
    end;
    then
A45: (downarrow a1) /\ (c.:([#]CompactSublatt L)) c= (downarrow a) /\ (c
    .: ([#]CompactSublatt L));
    ex_sup_of c.:(compactbelow a1),L by A8,WAYBEL_0:def 31;
    then
A46: a = "\/"(c.:(compactbelow a1),Imc) by A6,A7,A17,A16,WAYBEL_1:55;
    ex_sup_of c.:(compactbelow a1),Imc & ex_sup_of (downarrow a) /\ (c.:(
    [#] CompactSublatt L)),Imc by YELLOW_0:17;
    then a <= "\/"((downarrow a) /\ (c.:([#]CompactSublatt L)),Imc) by A46,A15
,A45,XBOOLE_1:1,YELLOW_0:34;
    then consider k be Element of Imc such that
A47: k in ((downarrow a) /\ (c.:([#]CompactSublatt L))) and
A48: a <= k by A5,A18,WAYBEL_3:def 1;
    k in downarrow a by A47,XBOOLE_0:def 4;
    then
A49: k <= a by WAYBEL_0:17;
    k in c.:([#]CompactSublatt L) by A47,XBOOLE_0:def 4;
    hence a9 in c.:([#]CompactSublatt L) by A48,A49,ORDERS_2:2;
  end;
  then
A50: [#]CompactSublatt Image c c= c.:([#]CompactSublatt L);
  c.:([#]CompactSublatt L) c= [#]CompactSublatt Image c by A1,Th23;
  hence thesis by A50,XBOOLE_0:def 10;
end;
