
theorem :: PROPOSITION 4.7 a)
  for L be algebraic LATTICE holds L is arithmetic iff CompactSublatt L
  is LATTICE
proof
  let L be algebraic LATTICE;
  thus L is arithmetic implies CompactSublatt L is LATTICE
  proof
    set x = the Element of L;
    assume
A1: L is arithmetic;
    compactbelow x is non empty by Def4;
    then consider z be object such that
A2: z in compactbelow x by XBOOLE_0:def 1;
    ex z9 be Element of L st z9 = z & x >= z9 & z9 is compact by A2;
    then CompactSublatt L is non empty join-inheriting meet-inheriting full
    SubRelStr of L by A1,Def1;
    hence thesis;
  end;
  assume
A3: CompactSublatt L is LATTICE;
  now
    let x,y be Element of L;
    assume that
A4: x in the carrier of CompactSublatt L and
A5: y in the carrier of CompactSublatt L and
    ex_inf_of {x,y},L;
    reconsider L9 = CompactSublatt L as non empty full SubRelStr of L by A4;
    reconsider x9 = x, y9 = y as Element of L9 by A4,A5;
    set X = compactbelow inf {x,y};
    reconsider c = "/\"({x,y},L9) as Element of L by YELLOW_0:58;
A6: inf {x,y} = sup X by Def3;
    X is non empty directed by Def4;
    then
A7: ex_sup_of X,L by WAYBEL_0:75;
A8: ex_inf_of {x9,y9},L9 by A3,YELLOW_0:21;
    then
A9: "/\"({x,y},L9) is_<=_than {x,y} by YELLOW_0:31;
    now
      let z be object;
      assume z in X;
      then consider z1 be Element of L such that
A10:  z = z1 and
A11:  inf {x,y} >= z1 and
A12:  z1 is compact;
A13:  z1 <= x "/\" y by A11,YELLOW_0:40;
      x "/\" y <= y by YELLOW_0:23;
      then z1 <= y by A13,ORDERS_2:3;
      then
A14:  z in compactbelow y by A10,A12;
      x "/\" y <= x by YELLOW_0:23;
      then z1 <= x by A13,ORDERS_2:3;
      then z in compactbelow x by A10,A12;
      hence z in compactbelow x /\ compactbelow y by A14,XBOOLE_0:def 4;
    end;
    then
A15: X c= compactbelow x /\ compactbelow y;
    now
      let b9 be Element of L9;
      reconsider b = b9 as Element of L by YELLOW_0:58;
      assume
A16:  b9 in X;
      then b9 in compactbelow y by A15,XBOOLE_0:def 4;
      then b <= y by Th4;
      then
A17:  b9 <= y9 by YELLOW_0:60;
      b9 in compactbelow x by A15,A16,XBOOLE_0:def 4;
      then b <= x by Th4;
      then b9 <= x9 by YELLOW_0:60;
      then b9 <= x9 "/\" y9 by A8,A17,YELLOW_0:19;
      hence b9 <= "/\"({x,y},L9) by A3,YELLOW_0:40;
    end;
    then
A18: X is_<=_than "/\"({x,y},L9) by LATTICE3:def 9;
    now
      let d be object;
      assume d in X;
      then
      ex d9 be Element of L st d9 = d & inf {x,y} >= d9 & d9 is compact;
      hence d in the carrier of L9 by Def1;
    end;
    then X c= the carrier of L9;
    then X is_<=_than c by A18,YELLOW_0:62;
    then
A19: sup X <= c by A7,YELLOW_0:30;
    y9 in {x,y} by TARSKI:def 2;
    then "/\"({x,y},L9) <= y9 by A9,LATTICE3:def 8;
    then
A20: c <= y by YELLOW_0:59;
    x9 in {x,y} by TARSKI:def 2;
    then "/\"({x,y},L9) <= x9 by A9,LATTICE3:def 8;
    then c <= x by YELLOW_0:59;
    then c <= x "/\" y by A20,YELLOW_0:23;
    then c <= sup X by A6,YELLOW_0:40;
    then c = sup X by A19,ORDERS_2:2;
    hence inf {x,y} in the carrier of CompactSublatt L by A6;
  end;
  then CompactSublatt L is meet-inheriting by YELLOW_0:def 16;
  hence thesis;
end;
