
theorem :: PROPOSITION 4.9.(i)
  for L be algebraic lower-bounded LATTICE for c be closure Function of
  L,L st c is directed-sups-preserving holds Image c is algebraic LATTICE
proof
  let L be algebraic lower-bounded LATTICE;
  let c be closure Function of L,L;
  assume
A1: c is directed-sups-preserving;
  c is idempotent by WAYBEL_1:def 13;
  then reconsider Imc = Image c as complete LATTICE by A1,YELLOW_2:35;
A2: now
    let x be Element of Imc;
    now
      let y,z be Element of Imc;
      assume that
A3:   y in compactbelow x and
A4:   z in compactbelow x;
      y is compact by A3,Th4;
      then
A5:   y << y by WAYBEL_3:def 2;
      z is compact by A4,Th4;
      then
A6:   z << z by WAYBEL_3:def 2;
      take v = y "\/" z;
      z <= v by YELLOW_0:22;
      then
A7:   z << v by A6,WAYBEL_3:2;
      y <= v by YELLOW_0:22;
      then y << v by A5,WAYBEL_3:2;
      then v << v by A7,WAYBEL_3:3;
      then
A8:   v is compact by WAYBEL_3:def 2;
      y <= x & z <= x by A3,A4,Th4;
      then v <= x by YELLOW_0:22;
      hence v in compactbelow x & y <= v & z <= v by A8,YELLOW_0:22;
    end;
    hence compactbelow x is non empty directed by WAYBEL_0:def 1;
  end;
  now
    let x be Element of Imc;
    consider y be Element of L such that
A9: c.y = x by YELLOW_2:10;
    sup compactbelow y in the carrier of L;
    then
A10: sup compactbelow y in dom c by FUNCT_2:def 1;
    compactbelow y is non empty directed by Def4;
    then
A11: c preserves_sup_of compactbelow y & ex_sup_of compactbelow y,L by A1,
WAYBEL_0:75,def 37;
    then c.sup compactbelow y = sup (c.:(compactbelow y)) by WAYBEL_0:def 31;
    then sup (c.:(compactbelow y)) in rng c by A10,FUNCT_1:def 3;
    then
A12: ex_sup_of (c.:(compactbelow y)),L & sup (c.:(compactbelow y)) in the
    carrier of Image c by YELLOW_0:17,def 15;
    for b be Element of Imc st b in compactbelow x holds b <= x by Th4;
    then x is_>=_than compactbelow x by LATTICE3:def 9;
    then
A13: sup compactbelow x <= x by YELLOW_0:32;
A14: c.:([#]CompactSublatt L) c= [#]CompactSublatt Image c by A1,Th23;
A15: c is monotone by A1,YELLOW_2:16;
    compactbelow y = downarrow y /\ [#]CompactSublatt L by Th5;
    then
A16: c.:(compactbelow y) c= c.:(downarrow y) /\ c.: ([#]CompactSublatt L)
    by RELAT_1:121;
    now
      let z be object;
      assume
A17:  z in c.:(compactbelow y);
      then consider v be object such that
A18:  v in dom c and
A19:  v in compactbelow y and
A20:  z = c.v by FUNCT_1:def 6;
      z in rng c by A18,A20,FUNCT_1:def 3;
      then reconsider z1 = z as Element of Imc by YELLOW_0:def 15;
      reconsider v as Element of L by A18;
      v <= y by A19,Th4;
      then c.v <= c.y by A15,WAYBEL_1:def 2;
      then
A21:  z1 <= x by A9,A20,YELLOW_0:60;
      z in c.:([#]CompactSublatt L) by A16,A17,XBOOLE_0:def 4;
      then z1 is compact by A14,Def1;
      hence z in compactbelow x by A21;
    end;
    then
A22: c.:(compactbelow y) c= compactbelow x;
    compactbelow x is directed by A2;
    then
A23: ex_sup_of compactbelow x,Imc by WAYBEL_0:75;
    c.:(compactbelow y) c= rng c by RELAT_1:111;
    then c.:(compactbelow y) is Subset of Imc by YELLOW_0:def 15;
    then
A24: ex_sup_of c.:(compactbelow y),Imc & sup (c.:(compactbelow y)) = "\/"(
    c.:( compactbelow y),Imc) by A12,YELLOW_0:64;
    c.y = c.sup compactbelow y by Def3
      .= sup (c.:(compactbelow y)) by A11,WAYBEL_0:def 31;
    then x <= sup compactbelow x by A9,A23,A24,A22,YELLOW_0:34;
    hence x = sup compactbelow x by A13,ORDERS_2:2;
  end;
  then Imc is satisfying_axiom_K;
  hence thesis by A2,Def4;
end;
